Hey there, future physicists! Welcome to a really important topic in current electricity: understanding cells, their Electromotive Force (EMF), and a little something called internal resistance. These concepts are the heart and soul of how batteries work and how we analyze electrical circuits.

Think about it: you want to power a light bulb, a phone, or a laptop. Where does the electrical "oomph" come from? It comes from a cell or a battery! Let's dive in and understand how these magical devices provide that push.

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### 1. What is a Cell (or a Battery)? - The Electrical "Pusher"!

Imagine you want water to flow through a pipe. Water won't flow unless there's a difference in pressure, right? You need a pump to lift the water to a higher level, creating that pressure difference.

Similarly, for electric current (which is just the flow of electric charge) to flow through a wire, you need something to create an electrical "pressure" difference. This "pressure" difference is what we call potential difference or voltage.

And that's precisely what a cell (or a battery, which is just a combination of multiple cells) does!
A cell is a device that converts chemical energy into electrical energy, creating and maintaining a potential difference across its terminals. This potential difference is what "pushes" charges around a circuit.

Think of it like this:

• Water Pump Analogy: A water pump lifts water from a lower reservoir to a higher one, creating a pressure difference that makes water flow through pipes.


• Electric Cell: An electric cell uses chemical reactions to push positive charges from a lower electric potential to a higher electric potential *inside* itself, creating a potential difference that makes current flow through wires.

Without a cell, charges would quickly spread out, and the potential difference would vanish, just like water eventually settles to the same level. A cell constantly works to maintain this potential difference.

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### 2. Electromotive Force (EMF - E) - The Cell's Full Potential

Now, when our water pump is running, there's a certain maximum pressure difference it *can* create, even if no water is flowing out yet. This is its full potential.

Similarly, an ideal cell has an inherent ability to create a potential difference. We call this the Electromotive Force (EMF), represented by the symbol E.

Key Idea: EMF is the maximum potential difference that a cell can provide across its terminals when no current is drawn from it. In other words, it's the potential difference when the circuit is "open."

Don't let the word "force" confuse you! EMF is NOT a force in the mechanical sense (like pushing a box). It's actually a measure of energy per unit charge.
Its unit is the Volt (V), just like potential difference. One volt means one joule of energy is supplied per coulomb of charge.

Mathematically, we can think of EMF as the work done by the cell (or any source of electrical energy) in moving a unit positive charge once around the complete circuit, including the internal part of the cell.

Why is it called "Electromotive Force" then? Historically, when these concepts were first being explored, scientists thought it was a "force" that drove charges. The name stuck, even though we now understand it's a potential difference or energy conversion per unit charge.

So, if you take a brand-new 1.5V AA battery and measure its voltage with a voltmeter *before* you connect it to anything, you're essentially measuring its EMF. It's the "advertised" voltage of the cell.

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### 3. Internal Resistance (r) - The Cell's Hidden Obstacle

Now, back to our water pump. Even the best pump isn't perfectly efficient. There's always some friction in its internal components, in the pipes leading to its outlet, or the pump itself consumes some energy to operate. This friction reduces the effective pressure available at the outlet compared to the maximum pressure the pump can generate.

In the same way, a real electric cell isn't perfect either. The materials inside the cell – the electrodes (metal plates) and the electrolyte (the chemical paste or liquid that allows charge movement) – all offer some opposition to the flow of charge *within the cell itself*. This opposition is called internal resistance, and we represent it with a lowercase 'r'.

Key Idea: Internal resistance (r) is the resistance offered by the cell's own materials (electrodes and electrolyte) to the flow of current within the cell.

Internal resistance means that when current flows through the cell, some of the cell's generated energy is wasted as heat *inside* the cell. This energy loss causes a drop in voltage within the cell itself.

Factors affecting internal resistance:

• Nature of electrolyte: Different chemicals offer different resistances.


• Concentration of electrolyte: Higher concentration usually means lower resistance up to a point.


• Distance between electrodes: Larger distance, more resistance.


• Area of electrodes dipping in electrolyte: Larger submerged area, lower resistance.


• Temperature: Generally, internal resistance decreases with increasing temperature.


• Age of cell: As a cell ages, its internal resistance often increases.

The unit of internal resistance is also the Ohm (Ω), just like external resistance.

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### 4. Terminal Voltage (V) - What You Actually Get!

So, the cell has an EMF (E), which is its full potential. But because of internal resistance (r), when current (I) actually flows through the circuit, there's a voltage drop *inside* the cell itself. This internal voltage drop is Ir (from Ohm's Law, V = IR, where 'R' here is 'r').

The voltage that you actually measure across the terminals of the cell *when current is being drawn from it* is called the Terminal Voltage (or Potential Difference), denoted by V.

The Fundamental Relationship: The terminal voltage (V) is the EMF (E) minus the voltage drop due to internal resistance (Ir).




V = E - Ir

Let's break this down:

• E: The total "push" provided by the cell (EMF).


• Ir: The "lost" voltage *inside* the cell due to its internal resistance when current I flows. This is often called the "lost volts".


• V: The actual "push" available to the external circuit (Terminal Voltage).

Think of our water pump again:
* E is the maximum height difference the pump can create.
* Ir is the height difference "lost" due to friction *within the pump and its pipes* when water is actually flowing.
* V is the actual height difference available at the outlet of the pump to drive water through the external pipes.

#### Important Scenarios:

1. Open Circuit (No current drawn):
* If no current (I = 0) is drawn from the cell (e.g., the circuit is open, or a voltmeter is connected directly across the terminals without any load), then the term 'Ir' becomes 0.
* In this case, V = E. The terminal voltage is equal to the EMF. This is how you measure a cell's EMF!

2. Closed Circuit (Current drawn):
* When current (I > 0) is drawn from the cell, then 'Ir' will be a positive value.
* Therefore, V < E. The terminal voltage will always be less than the EMF. The cell is "working" and some energy is dissipated internally.

3. Short Circuit (High current drawn):
* If you connect a wire of negligible resistance directly across the terminals, you create a "short circuit."
* The current drawn would be very high (theoretically, I = E/r if only internal resistance exists).
* This is dangerous as it can overheat the cell and drain it very quickly. The terminal voltage V would approach zero as the current becomes very large.

#### JEE/CBSE Focus:
This relationship, V = E - Ir, is extremely crucial! It's the cornerstone for analyzing circuits containing real batteries. You'll use it to calculate currents, potential differences across external components, and even the internal resistance itself. Many problems will involve applying Ohm's Law (V=IR) to the external circuit and then connecting it back to the cell's EMF and internal resistance.

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### Let's Summarize the Fundamentals:

Concept	Description	Symbol / Unit	Analogy
Cell / Battery	A device that converts chemical energy into electrical energy to maintain potential difference.	-	Water pump
EMF	The maximum potential difference a cell can provide when no current is drawn (open circuit). It's the total energy supplied per unit charge.	E (Volts, V)	Max height difference pump can create
Internal Resistance	The opposition to current flow offered by the cell's own materials (electrodes & electrolyte).	r (Ohms, Ω)	Friction inside the pump/pipes
Terminal Voltage	The actual potential difference available across the cell's terminals when current is being drawn from it.	V (Volts, V)	Actual height difference at pump outlet
Key Formula	V = E - Ir (for discharge)

Understanding these basic concepts thoroughly will make your journey through more complex DC circuits much smoother. Always remember that a real cell is not just an EMF source, but an EMF source with a tiny resistor hidden inside it!

Welcome, future engineers! Today, we're diving deep into the heart of any DC circuit – the cell, its Electromotive Force (EMF), and its inherent internal resistance. These are fundamental concepts, crucial not just for understanding basic circuits but also for tackling complex problems in JEE Main & Advanced. So, let's build this understanding brick by brick.

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### Introduction: The Purpose of a Cell

At its core, a cell (or battery, which is a combination of cells) is an energy converter. It converts chemical energy stored within it into electrical energy, thereby maintaining a potential difference across its terminals and driving electric current through an external circuit. Think of it as the 'heart' of an electrical circuit, constantly pumping charge and providing the necessary energy for devices to operate.

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### 1. Electromotive Force (EMF) – The Engine of the Circuit

Imagine you have a water pump that lifts water to a certain height. The "potential energy" imparted to the water is analogous to the potential difference a cell creates.

#### 1.1 Defining EMF (E or ℰ)

The Electromotive Force (EMF) of a cell is defined as the maximum potential difference that the cell can provide across its terminals when no current is drawn from it (i.e., in an open circuit). More fundamentally, EMF represents the work done by the cell (or non-conservative forces within the cell) in moving a unit positive charge from its low potential terminal to its high potential terminal (or around the complete circuit).

* Source of EMF: Inside a cell, chemical reactions separate positive and negative charges, creating a potential difference. These chemical forces are non-conservative, meaning the work they do depends only on the initial and final states, not the path.
* Units: Like potential difference, EMF is measured in Volts (V), which is Joules per Coulomb (J/C).
* Symbol: Often denoted by `E` or `ℰ`.

#### 1.2 Why "Force" but not a Force?

This is a common point of confusion. Despite its name, EMF is NOT a force in the Newtonian sense (measured in Newtons). It is a measure of energy per unit charge, hence it's a potential difference. The term "force" in EMF harks back to an older understanding of electricity. It's more accurately described as an "electromotive potential" or "electromotive voltage."

#### 1.3 Ideal vs. Real Cells

* An ideal cell is a theoretical concept where the EMF is always equal to the terminal voltage, regardless of the current drawn. It has no internal energy loss.
* A real cell, however, always has some inherent resistance to the flow of charge within its own structure. This brings us to internal resistance.

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### 2. Internal Resistance (r) – The Cell's Hidden Opponent

A real cell is not just a source of EMF; it also possesses an internal resistance.

#### 2.1 Origin of Internal Resistance

The internal resistance (r) of a cell is the resistance offered by the electrolyte and the electrodes within the cell to the flow of electric current. Imagine current flowing through the chemical solution and metal plates inside the battery – they don't offer zero resistance.

Several factors influence internal resistance:
* Nature of electrolyte: Different chemical compositions offer varying resistance.
* Concentration of electrolyte: Higher concentration generally means lower internal resistance.
* Distance between electrodes: Smaller distance, lower resistance.
* Area of electrodes immersed in electrolyte: Larger area, lower resistance.
* Temperature: Usually, higher temperature leads to lower internal resistance.
* Age of the cell: As a cell ages, its internal resistance tends to increase.

#### 2.2 Impact on Terminal Voltage

Because of internal resistance, some of the EMF produced by the cell is "lost" or dissipated as heat within the cell itself when current flows. This means the potential difference available across the external terminals of the cell (terminal voltage, V) will always be less than its EMF when it is actively supplying current to a circuit.

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### 3. The Fundamental Equation: Connecting EMF, Terminal Voltage, and Internal Resistance

This is where all these concepts come together.

#### 3.1 Derivation

Consider a real cell with EMF `E` and internal resistance `r`. Let it be connected to an external resistance `R` (this could be a bulb, a resistor, etc.) through which a current `I` flows.

When current `I` flows through the cell:
1. The cell generates an EMF `E`.
2. The current `I` passes through the internal resistance `r` of the cell.
3. According to Ohm's Law, there is a potential drop across the internal resistance, which is `V_internal = I * r`. This potential drop represents the energy lost inside the cell.
4. The potential difference available across the terminals of the cell, which is the terminal voltage (V), is therefore the EMF minus this internal potential drop.

So, the fundamental equation is:



V = E - I r



Here:
* `V` is the terminal voltage across the cell (the voltage measured across its external terminals).
* `E` is the EMF of the cell.
* `I` is the current flowing out of the cell into the external circuit.
* `r` is the internal resistance of the cell.

#### 3.2 Understanding the Equation: V = E - Ir

This equation is a cornerstone of DC circuits. It explains why a battery's voltage "drops" under load. The `Ir` term is often called the "lost voltage" or "internal voltage drop".

We can also relate `V` to the external circuit: if `R_ext` is the equivalent external resistance, then `V = I * R_ext`.
Substituting this into the main equation:
`I * R_ext = E - I * r`
`I * (R_ext + r) = E`



I = E / (R_ext + r)



This formula provides the total current in a simple series circuit comprising an EMF source, its internal resistance, and an external resistance. It's essentially Ohm's Law for the entire circuit.

#### 3.3 Case Studies

Let's analyze different scenarios:

1. Open Circuit (No Current Drawn):
* When the cell is not connected to any external circuit, or the circuit is broken, no current flows (`I = 0`).
* From `V = E - Ir`, if `I = 0`, then `V = E - 0 * r`, so V = E.
* Insight: The terminal voltage in an open circuit is equal to the cell's EMF. This is how you would ideally measure the EMF of a cell.

2. Closed Circuit (Discharging):
* When the cell is connected to an external resistance `R_ext` and current `I` flows out of its positive terminal (`I > 0`).
* `V = E - Ir`. Since `I` and `r` are positive, `Ir` is positive.
* Insight: The terminal voltage `V` will always be less than the EMF (E). This is the normal operating condition for a cell powering a device. The larger the current drawn, the lower the terminal voltage.

3. Short Circuit:
* If the terminals of the cell are directly connected without any external resistance (or `R_ext = 0`), this is a short circuit.
* In this extreme case, `V = I * R_ext = I * 0 = 0`.
* Using `V = E - Ir`, we get `0 = E - I_sc * r`.
* The short-circuit current is I_sc = E / r.
* Warning: Short circuits are dangerous as they draw a very large current, leading to rapid energy dissipation and potential damage or explosion of the cell due to excessive heat.

4. Charging a Cell (Crucial for JEE Advanced):
* This scenario occurs when an external source (e.g., a charger) forces current *into* the positive terminal of the cell, effectively reversing the chemical reactions. In this case, the cell acts as a load, not a source.
* If current `I` flows *into* the positive terminal (opposite to discharge), the equation modifies. The external charger needs to overcome both the cell's EMF `E` and the potential drop `Ir` across its internal resistance.
* Therefore, the terminal voltage `V` across the cell during charging must be V = E + Ir.
* Insight: During charging, the terminal voltage `V` is greater than its EMF (E).

Condition	Current Direction	Terminal Voltage (V)	Relationship to EMF (E)
Open Circuit	I = 0	V = E	V = E
Discharging	I flows out of positive terminal	V = E - Ir	V < E
Charging	I flows into positive terminal	V = E + Ir	V > E
Short Circuit	I = E/r (max current)	V = 0	V < E (extreme case)

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### 4. Power Dynamics in a Circuit with a Real Cell

Understanding power flow is vital for circuit analysis.

#### 4.1 Total Power Generated by EMF

The total electrical power generated by the non-conservative forces (chemical reactions) within the cell is given by:



P_total = E * I



This is the rate at which chemical energy is converted into electrical energy by the cell.

#### 4.2 Power Dissipated Internally

A portion of this total power is always dissipated as heat within the cell itself due to its internal resistance `r`. This is the "lost power":



P_internal = I^2 * r




#### 4.3 Power Delivered to External Circuit

The useful power delivered to the external resistance `R_ext` (where work is done, e.g., a bulb glows, a motor runs) is:



P_external = V * I = I^2 * R_ext = V^2 / R_ext



By conservation of energy, the total power generated must equal the sum of power dissipated internally and power delivered externally:
`P_total = P_internal + P_external`
`E * I = I^2 * r + I^2 * R_ext`
`E * I = I * (I * r + I * R_ext)`
`E = I * r + I * R_ext`
`E = Ir + V`
`V = E - Ir` (This beautifully re-derives our fundamental voltage equation from power considerations!)

#### 4.4 Maximum Power Transfer Theorem (JEE Advanced Focus)

For a given cell with EMF `E` and internal resistance `r`, the power delivered to the external resistance `R_ext` is:
`P_external = I^2 * R_ext = (E / (R_ext + r))^2 * R_ext`

To find the condition for maximum power transfer to the external load, we differentiate `P_external` with respect to `R_ext` and set it to zero (`dP_external / dR_ext = 0`).

After performing the differentiation (using the quotient rule), you will find that the power delivered to the external circuit is maximum when:



R_ext = r



Insight: This is a critical result for JEE Advanced. Maximum power is delivered to the load when the external resistance matches the internal resistance of the source. At this point, the efficiency is 50% (half the generated power is lost internally).

The maximum power transferred is:
`P_max = (E / (r + r))^2 * r = (E / 2r)^2 * r = E^2 / (4r^2) * r`



P_max = E^2 / (4r)




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### 5. Graphical Analysis: V vs. I Characteristics

The equation `V = E - Ir` is of the form `y = c - mx`, which represents a straight line.
* `V` is on the y-axis.
* `I` is on the x-axis.
* `E` is the y-intercept (when `I = 0`, `V = E`).
* `-r` is the slope of the line.

(Self-correction: Cannot generate images, so will describe it clearly.)

Imagine a graph with Current (I) on the horizontal axis and Terminal Voltage (V) on the vertical axis.
1. Y-intercept: When `I = 0` (open circuit), `V = E`. So, the graph starts at `(0, E)` on the V-axis.
2. Slope: As current `I` increases, the terminal voltage `V` decreases linearly because of the `Ir` drop. The slope of this line is `-r`.
3. X-intercept: When `V = 0` (short circuit), `I = E/r`. So, the graph intercepts the I-axis at `(E/r, 0)`.

This linear relationship is often used in labs to determine the EMF and internal resistance of a cell by plotting V vs I and finding the intercept and slope.

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### 6. Illustrative Examples

Let's solidify our understanding with some practical examples.

#### Example 1: Calculating Terminal Voltage

A cell has an EMF of 12 V and an internal resistance of 0.5 Ω. What is the terminal voltage of the cell when it supplies a current of 2 A to an external circuit?

Solution:
Given:
* EMF, `E = 12 V`
* Internal resistance, `r = 0.5 Ω`
* Current, `I = 2 A`

Using the formula `V = E - Ir`:
`V = 12 V - (2 A * 0.5 Ω)`
`V = 12 V - 1 V`
`V = 11 V`

The terminal voltage of the cell is 11 V. This shows that 1V is lost internally due to the cell's resistance.

#### Example 2: Determining Internal Resistance

A cell has an EMF of 6 V. When connected to an external resistance of 2.5 Ω, the current flowing through the circuit is 2 A. Calculate the internal resistance of the cell.

Solution:
Given:
* EMF, `E = 6 V`
* External resistance, `R_ext = 2.5 Ω`
* Current, `I = 2 A`

We know the formula for current in a circuit: `I = E / (R_ext + r)`
Rearranging to find `r`:
`I * (R_ext + r) = E`
`R_ext + r = E / I`
`r = (E / I) - R_ext`

Substitute the values:
`r = (6 V / 2 A) - 2.5 Ω`
`r = 3 Ω - 2.5 Ω`
`r = 0.5 Ω`

The internal resistance of the cell is 0.5 Ω.

#### Example 3: Maximum Power Transfer

A battery has an EMF of 10 V and an internal resistance of 2 Ω. What external resistance should be connected to the battery to draw maximum power from it? Also, calculate this maximum power.

Solution:
Given:
* EMF, `E = 10 V`
* Internal resistance, `r = 2 Ω`

For maximum power transfer, the external resistance must be equal to the internal resistance:
`R_ext = r`
`R_ext = 2 Ω`

The external resistance for maximum power transfer is 2 Ω.

Now, calculate the maximum power:
`P_max = E^2 / (4r)`
`P_max = (10 V)^2 / (4 * 2 Ω)`
`P_max = 100 / 8 W`
`P_max = 12.5 W`

The maximum power transferred to the external circuit is 12.5 W.

#### Example 4: Cell Charging Scenario

A 12 V car battery with an internal resistance of 0.1 Ω is being charged by an external supply. If the charging current is 10 A, what is the terminal voltage across the battery during charging?

Solution:
Given:
* EMF of the battery, `E = 12 V`
* Internal resistance, `r = 0.1 Ω`
* Charging current, `I = 10 A` (current is forced into the positive terminal)

For charging, the terminal voltage is given by `V = E + Ir`:
`V = 12 V + (10 A * 0.1 Ω)`
`V = 12 V + 1 V`
`V = 13 V`

The terminal voltage across the battery during charging is 13 V. Notice how it's higher than the battery's EMF, as the external charger must push against both the EMF and the internal resistance.

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### JEE Mains & Advanced Takeaways

* Understanding `V = E - Ir` (and `V = E + Ir`): This is the most crucial formula. Be adept at applying it for discharging and charging scenarios.
* Open vs. Closed Circuits: Differentiate between EMF (open circuit voltage) and terminal voltage (closed circuit voltage).
* Maximum Power Transfer: This is a frequently tested concept, especially in JEE Advanced. Remember `R_ext = r` and `P_max = E^2 / (4r)`.
* Graphical Analysis: Interpreting `V` vs `I` graphs to find `E` (y-intercept) and `r` (negative of the slope) is a common question type.
* Practical Implications: Internal resistance limits the current a battery can supply and reduces its efficiency. Understanding it is key to designing efficient power systems.

By mastering these concepts, you'll have a robust foundation for tackling more complex DC circuit problems, including combinations of cells and more intricate network analysis. Keep practicing with diverse problems, and you'll find these ideas becoming second nature!

Navigating the concepts of EMF, internal resistance, and terminal voltage is crucial for excelling in Current Electricity. These mnemonics and shortcuts are designed to help you quickly recall key formulas and distinctions, especially under exam pressure.

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🧠 Mnemonics for Core Concepts

• Distinguishing EMF (E) and Terminal Voltage (V)
  
  
  
  • EMF (E): Think "Empty Circuit" (Open Circuit).
    
    
    
    • This is the maximum potential difference the cell can provide when no current is drawn (external circuit is open).
    
    
    
    
  
  
  • Terminal Voltage (V): Think "Very Busy Circuit" (Closed Circuit).
    
    
    
    • This is the actual potential difference across the cell terminals when current is flowing through the external circuit. It's always less than or equal to EMF.
    
    
    
    
  
  
  
  



• Formula for Terminal Voltage (Discharging Cell): V = E - Ir
  
  
  
  • Mnemonic: "Voltage External Is Reduced"
    
    
    
    • V: Terminal Voltage (what you get at the terminals)
    
    
    • E: EMF (the cell's inherent potential)
    
    
    • I: Current flowing through the circuit
    
    
    • r: Internal resistance of the cell
    
    
    • The word "Reduced" directly implies the minus sign, reminding you that the terminal voltage is EMF minus the voltage drop across the internal resistance (Ir). This is the most common scenario for a cell powering a circuit.
    
    
    
    
  
  
  
  



• Ohm's Law for the Entire Circuit: I = E / (R + r)
  
  
  
  • Mnemonic: "In Every Room, Resistance is Plus"
    
    
    
    • I: Total current in the circuit
    
    
    • E: Total EMF of the cell
    
    
    • R: External Resistance (load)
    
    
    • r: Internal Resistance of the cell
    
    
    • This helps you remember that to find the total current, you divide the total EMF by the sum of all resistances in the path – both external and internal.
    
    
    
    
  
  
  
  



• Condition for V = E (Terminal Voltage equals EMF)
  
  
  
  • Mnemonic: "Very Easy When Is 0 or r is 0"
    
    
    
    • Terminal voltage V equals EMF E only under two specific conditions:
      
      
      
      1. When no current is drawn (I = 0), i.e., the circuit is open.
      
      
      2. When the cell is ideal, meaning its internal resistance (r = 0).
      
      
      
      
    
    
    
    
  
  
  
  

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⚡ Exam Shortcuts & Quick Checks

• Charging vs. Discharging Cells:
  
  
  
  • Discharging (most common): Cell provides current. V = E - Ir. Terminal voltage is less than EMF.
  
  
  • Charging: External source forces current into the cell. V = E + Ir. Terminal voltage is greater than EMF.
  
  
  • Quick Check: If the cell is supplying power, EMF is 'losing' voltage internally. If it's being charged, it's 'gaining' voltage from the external source, plus the internal drop.
  
  
  
  



• Graphical Interpretation (V-I Graph):
  
  
  
  • For a cell discharging, the equation V = -rI + E is a straight line of the form y = mx + c.
    
    
    
    • Plotting V on the y-axis and I on the x-axis:
      
      
      
      • The Y-intercept gives the EMF (E).
      
      
      • The magnitude of the slope gives the internal resistance (r). Note: The slope will be negative.
      
      
      
      
    
    
    
    
  
  
  
  

Example Mental Check: If a cell has an EMF of 1.5 V and internal resistance of 0.1 Ω, and draws 2 A current:

• Using "Voltage External Is Reduced": V = 1.5 - (2 * 0.1) = 1.5 - 0.2 = 1.3 V. The terminal voltage is indeed reduced.

Mastering these fundamental relationships and using these simple aids will significantly boost your confidence and speed in solving problems related to cells, EMF, and internal resistance. Keep practicing!

Mastering the concepts of EMF, internal resistance, and terminal voltage is crucial for excelling in Current Electricity problems for both JEE and CBSE exams. Here are some quick tips to help you solidify your understanding and tackle related questions effectively.

Quick Tips: Cell, EMF and Internal Resistance

• Distinguish EMF (E) from Terminal Voltage (V):
  
  
  
  • EMF (Electromotive Force): This is the maximum potential difference a cell can provide. It's the work done per unit charge by the non-conservative forces (chemical forces) within the cell. EMF is a characteristic of the cell itself and remains constant unless the cell's chemical composition changes.
  
  
  • Terminal Voltage: This is the actual potential difference across the cell's terminals when it is delivering current to an external circuit. Terminal voltage is always less than or equal to the EMF due to the voltage drop across the cell's internal resistance.
  
  
  
  


• Internal Resistance (r): The "Hidden" Resistor:
  
  
  
  • Every real cell has an internal resistance, which arises from the resistance offered by the electrolyte and electrodes within the cell.
  
  
  • This internal resistance causes a voltage drop (Ir) within the cell itself when current (I) flows.
  
  
  • Key Relationship: V = E - Ir. This is the most fundamental equation.
    
    
    
    • If I = 0 (open circuit), then V = E. This is how EMF is measured.
    
    
    • If I > 0 (cell discharging), then V < E.
    
    
    • If the cell is being charged (current flowing into the positive terminal), then V = E + Ir, and V > E.
    
    
    
    
  
  
  
  


• Current in the Circuit:
  
  
  
  • When a cell with EMF 'E' and internal resistance 'r' is connected to an external resistance 'R', the total resistance in the circuit is (R + r).
  
  
  • Therefore, the current flowing through the circuit is I = E / (R + r).
  
  
  • You can also write Terminal Voltage as V = IR (voltage across the external resistor).
  
  
  
  


• Power Considerations:
  
  
  
  • Total Power supplied by cell: P_total = EI = I²(R + r)
  
  
  • Power dissipated in external resistance (useful power): P_ext = I²R = V²/R = VI
  
  
  • Power dissipated internally (wasted as heat): P_internal = I²r
  
  
  • Check: P_total = P_ext + P_internal. This energy conservation principle is crucial for problem-solving.
  
  
  
  


• Maximizing Power Transfer (JEE Specific):
  
  
  
  • A common JEE problem type involves finding the external resistance (R) for which maximum power is delivered by the cell.
  
  
  • Maximum power is delivered to the external resistance when the external resistance equals the internal resistance of the source: R = r.
  
  
  • At maximum power transfer, P_max = E² / (4r).
  
  
  
  


• Factors Affecting Internal Resistance (CBSE Conceptual):
  
  
  
  • Nature of electrolyte (concentration, conductivity).
  
  
  • Nature of electrodes.
  
  
  • Distance between electrodes.
  
  
  • Temperature (usually decreases with increasing temperature).
  
  
  • Age of the cell (internal resistance increases as the cell ages).
  
  
  
  

Exam Strategy: Always draw a clear circuit diagram, label EMF, internal resistance, and external resistance. Systematically apply Kirchhoff's laws or the basic formulas to find unknown quantities. For conceptual questions, especially differentiate between ideal conditions (r=0) and real-world scenarios (r>0).

Intuitive Understanding: Cell, EMF, and Internal Resistance

Understanding the behavior of a real battery or cell is crucial for both board exams and competitive tests like JEE. Unlike ideal voltage sources, real cells have limitations, which we model using the concepts of Electromotive Force (EMF) and Internal Resistance.

Imagine a water pump system:

* The pump's job is to lift water from a lower level to a higher level, giving it potential energy. This "lifting capacity" is inherent to the pump.
* However, even the best pump has some internal friction or resistance to water flow within its own mechanism.

Let's apply this analogy to a cell:

1. Electromotive Force (EMF or E)

• 
  What it is: EMF is the maximum potential difference a cell can provide when no current is being drawn from it (i.e., when the circuit is open). It's the total "push" or energy per unit charge that the chemical reactions inside the cell are capable of generating.
  


• 
  Analogy: In our water pump analogy, EMF is like the maximum height the pump *could* lift water to, assuming no internal friction or losses within the pump itself. It represents the inherent "power" of the chemical reactions.
  


• 
  Key Idea: EMF is an intrinsic property of the cell's chemistry. It's the cause, not the effect, of potential difference.
  


• 
  JEE Tip: EMF is always constant for a given cell, irrespective of the current drawn or the external resistance.
  

2. Internal Resistance (r)

• 
  What it is: This is the resistance offered by the electrolyte and electrodes within the cell itself to the flow of electric current. It's an unavoidable part of any real cell.
  


• 
  Analogy: This is like the internal friction or resistance that the water encounters *within the pump's casing* as it's being lifted. This internal friction causes some energy to be lost as heat.
  


• 
  Consequence: When current (I) flows through the cell, a portion of the EMF is "lost" or consumed within the cell due to this internal resistance. This voltage drop inside the cell is given by I * r.
  


• 
  Important for JEE & CBSE: Internal resistance leads to a reduction in the voltage available to the external circuit. Factors affecting 'r' include electrode area, distance between electrodes, and electrolyte concentration.
  

3. Terminal Potential Difference (V)

• 
  What it is: This is the actual potential difference available across the terminals of the cell when current is being drawn from it (i.e., when the circuit is closed). It's the voltage supplied to the external circuit.
  


• 
  Analogy: This is the actual pressure difference or height of water that is available *at the outlet pipe* of the pump, *after* accounting for the internal friction losses within the pump itself.
  


• 
  Relationship: The terminal potential difference (V) is the EMF (E) minus the voltage drop across the internal resistance (Ir).
  

  V = E - I r

  
  When a cell is discharging (supplying current), its terminal voltage is always less than its EMF.
  


• 
  Key takeaway: EMF is what the cell *could* provide, while Terminal Voltage is what it *actually does* provide to the external circuit under load.
  

In summary, a real cell is like an ideal voltage source (EMF) connected in series with a small resistor (internal resistance) *inside* the cell. This internal resistance "eats up" some of the cell's potential, leaving a reduced terminal voltage for the rest of the circuit. Understanding this distinction is fundamental for solving problems related to DC circuits with real batteries.

Understanding Cell, EMF, and Internal Resistance is fundamental to DC circuits. Several other topics are intricately linked, building the necessary foundation or extending its applications. Mastering these related concepts will significantly enhance your ability to solve complex circuit problems in both board exams and competitive exams like JEE Main.

Here are the key related topics:

• 
  Electric Current and Potential Difference:
  
  
  
  • Relation: These are the most basic prerequisites. A cell's EMF drives the current, and potential difference (terminal voltage) is the work done per unit charge between its terminals. Internal resistance influences this potential difference.
  
  
  • JEE/CBSE Relevance: Core definitions and understanding of how charge flows and energy is transferred are essential for all circuit analysis.
  
  
  
  


• 
  Ohm's Law (V = IR):
  
  
  
  • Relation: Ohm's Law is crucial for understanding the voltage drop across any external resistance in a circuit, as well as the voltage drop across the internal resistance (Ir). The terminal voltage of a cell (V) is directly related to its EMF (E) and internal resistance (r) through the equation V = E - Ir.
  
  
  • JEE/CBSE Relevance: Absolutely critical for solving any circuit problem involving resistors and cells.
  
  
  
  


• 
  Resistors in Series and Parallel Combinations:
  
  
  
  • Relation: Often, cells (along with their internal resistances) are part of circuits with multiple external resistors. Understanding how to calculate equivalent resistance for series and parallel combinations is vital for finding the total current drawn from the cell and, consequently, its terminal voltage.
  
  
  • JEE/CBSE Relevance: A fundamental skill required for simplifying circuits and applying Ohm's Law effectively.
  
  
  
  


• 
  Kirchhoff's Laws (KCL and KVL):
  
  
  
  • Relation: While Ohm's Law is sufficient for simple circuits, complex circuits with multiple cells and branches necessitate Kirchhoff's Laws. Kirchhoff's Voltage Law (KVL) is particularly important as it states that the algebraic sum of EMFs and voltage drops (IR products, including Ir for internal resistance) around any closed loop must be zero.
  
  
  • JEE Relevance: Highly important for JEE Advanced and complex JEE Main problems involving multiple cells or sources. CBSE also covers this, but usually with simpler applications.
  
  
  
  


• 
  Electrical Power and Energy:
  
  
  
  • Relation: A cell is a source of electrical energy. Understanding concepts like power delivered by the cell (EI), power dissipated in the external circuit (I²R), and power dissipated internally (I²r) due to internal resistance, are crucial. The efficiency of a cell is often related to the ratio of power delivered externally to the total power generated.
  
  
  • JEE/CBSE Relevance: Essential for quantitative analysis of circuit performance and energy conservation.
  
  
  
  


• 
  Potentiometer:
  
  
  
  • Relation: The potentiometer is an important instrument specifically designed to measure the EMF of a cell without drawing any current (thus avoiding the effect of internal resistance) and also to compare EMFs of two cells, and precisely measure the internal resistance of a cell.
  
  
  • JEE/CBSE Relevance: A significant topic for both JEE and CBSE, often appearing as experimental applications. Questions typically involve balancing length and calculations related to EMF and internal resistance.
  
  
  
  

Mastering these interconnected topics ensures a holistic understanding of how cells function in electric circuits, a frequently tested area in competitive exams.

📌 Common Exam Traps: Cell, EMF and Internal Resistance

Understanding the nuances of cells, electromotive force (EMF), and internal resistance is crucial for success in both board exams and JEE. Students often fall into specific traps due to conceptual misunderstandings or oversight. Be vigilant against these common pitfalls:

• 
  Trap 1: Confusing EMF (E) with Terminal Voltage (V)
  
  
  
  • The Error: Many students incorrectly use EMF (E) and terminal voltage (V) interchangeably.
  
  
  • The Clarification:
    
    
    
    • EMF (E) is the maximum potential difference a cell can provide when no current is drawn from it (open circuit). It's a characteristic of the cell itself.
    
    
    • Terminal Voltage (V) is the actual potential difference across the cell's terminals when current (I) is flowing through the external circuit.
    
    
    
    
  
  
  • Key Distinction: In a real cell, due to internal resistance (r), V is always less than E when the cell is discharging: V = E - Ir. Only in an ideal cell (r=0) or an open circuit (I=0) does V = E.
  
  
  
  


• 
  Trap 2: Incorrect Application of Charging vs. Discharging Formulas
  
  
  
  • The Error: Students often use V = E - Ir even when the cell is being charged.
  
  
  • The Clarification:
    
    
    
    • Use V = E - Ir when the cell is discharging (current flows out from the positive terminal to the external circuit).
    
    
    • Use V = E + Ir when the cell is charging (current is forced into the positive terminal by an external source).
    
    
    
    
  
  
  • Think Direction: If current *leaves* the positive terminal, it's discharging. If current *enters* the positive terminal, it's charging.
  
  
  
  


• 
  Trap 3: Ignoring Internal Resistance in Circuit Calculations
  
  
  
  • The Error: Treating all cells as ideal (r=0) unless explicitly stated, or forgetting to include 'r' in total circuit resistance.
  
  
  • The Clarification: In most practical problems (especially in JEE), cells are non-ideal and possess internal resistance. It must be accounted for as an additional resistance in series with the cell's EMF.
  
  
  • Impact: The total resistance of the circuit will be R_total = R_external + R_internal (where R_internal is the equivalent internal resistance of all cells).
  
  
  
  


• 
  Trap 4: Misinterpreting Open Circuit and Short Circuit Conditions
  
  
  
  • The Error: Confusing the conditions under which EMF is measured versus maximum current is drawn.
  
  
  • The Clarification:
    
    
    
    • Open Circuit: No external connection, so no current flows (I=0). Here, the terminal voltage is equal to the EMF (V = E). This is how EMF is experimentally determined.
    
    
    • Short Circuit: The terminals are connected directly with negligible external resistance (R=0). The current becomes maximum: I_max = E/r. In this case, the terminal voltage V = E - I_maxr = E - (E/r)r = 0.
    
    
    
    
  
  
  • Remember: An open circuit measures E; a short circuit yields I_max and V=0.
  
  
  
  


• 
  Trap 5: Incorrect Power Calculations
  
  
  
  • The Error: Mixing up power generated by the cell, power delivered to the external circuit, and power dissipated internally.
  
  
  • The Clarification:
    
    
    
    • Total Power Generated by Cell: P_total = E * I
    
    
    • Power Delivered to External Circuit: P_external = V * I = I²R_external = V²/R_external
    
    
    • Power Dissipated Internally: P_internal = I²r
    
    
    
    
  
  
  • Energy Conservation: P_total = P_external + P_internal. Always account for internal dissipation.
  
  
  
  


• 
  Trap 6: Complex Cell Combinations (JEE Specific)
  
  
  
  • The Error: Forgetting the rules for combining cells, especially when they are non-identical or have different polarities.
  
  
  • The Clarification:
    
    
    
    • Series: E_eq = ΣE_i (with signs based on polarity), r_eq = Σr_i.
    
    
    • Parallel (identical cells): E_eq = E, r_eq = r/n.
    
    
    • Parallel (non-identical cells): This requires a more advanced formula or Kirchhoff's laws: E_eq = (ΣE_i/r_i) / (Σ1/r_i) and 1/r_eq = Σ1/r_i. Students often struggle with this formula or its application.
    
    
    
    
  
  
  • JEE Focus: Be prepared for mixed combinations of cells with varying EMFs and internal resistances, often requiring Kirchhoff's Laws for precise analysis.
  
  
  
  

💪 Pro Tip: Always draw clear circuit diagrams and label EMFs, internal resistances, and current directions carefully to avoid these common traps. Practice problems with non-ideal cells extensively!

⚡ Key Takeaways: Cell, EMF and Internal Resistance ⚡

Understanding the concepts of Electromotive Force (EMF), Terminal Potential Difference (V), and Internal Resistance (r) is fundamental for solving DC circuit problems in both board exams and JEE. These concepts define the actual output characteristics of a real battery or cell.

• 
  Electromotive Force (EMF, E):
  
  
  
  • Definition: The work done by the cell (or energy supplied by the cell) per unit charge to move the charge from the lower potential terminal to the higher potential terminal inside the cell. It represents the maximum potential difference a cell can provide.
  
  
  • Characteristic: EMF is the potential difference across the cell terminals when no current is drawn from it (i.e., in an open circuit, external resistance R → ∞).
  
  
  • Unit: Volts (V).
  
  
  
  


• 
  Internal Resistance (r):
  
  
  
  • Definition: The resistance offered by the electrolyte and electrodes within the cell to the flow of current.
  
  
  • Impact: It causes a voltage drop within the cell itself when current flows, leading to the terminal potential difference being less than the EMF.
  
  
  • Factors affecting 'r' (JEE & Boards):
    
    
    
    • Concentration of electrolyte (higher concentration usually means lower 'r').
    
    
    • Nature of electrodes.
    
    
    • Distance between electrodes (larger distance means higher 'r').
    
    
    • Area of electrodes immersed in electrolyte (larger area means lower 'r').
    
    
    • Temperature (higher temperature usually means lower 'r').
    
    
    
    
  
  
  • Unit: Ohms (Ω).
  
  
  
  


• 
  Terminal Potential Difference (V):
  
  
  
  • Definition: The actual potential difference across the terminals of a cell when it is supplying current to an external circuit.
  
  
  • Relationship during Discharge (JEE & Boards): When a cell is discharging (supplying current 'I' to an external resistance 'R'), the voltage drop across the internal resistance is Ir. Thus, the terminal potential difference is:
    

    V = E - Ir

    
    Alternatively, since V = IR (across the external resistor), we have I = E / (R + r). Substituting this into V = IR gives V = E * R / (R + r).
    
  
  
  • Relationship during Charging: When a cell is being charged (current 'I' enters the positive terminal), the terminal potential difference becomes:
    

    V = E + Ir

    
    
  
  
  • Key Distinction: V ≤ E when discharging, and V ≥ E when charging. In an ideal cell (r=0), V = E.
  
  
  
  


• 
  Power Considerations (JEE Specific):
  
  
  
  • Power delivered to external circuit: P_ext = VI = I²R = V² / R
  
  
  • Power dissipated inside the cell (due to internal resistance): P_int = I²r
  
  
  • Total power generated by the cell: P_total = EI
  
  
  • Efficiency of the cell: η = P_ext / P_total = (I²R) / (EI) = R / (R + r)
  
  
  • Maximum Power Transfer Theorem (JEE Advanced): Power delivered to the external load is maximum when the external resistance R = r. The maximum power is then P_max = E² / (4r).
  
  
  
  

Mastering these relationships is crucial. Pay special attention to the conditions (discharging vs. charging, open vs. closed circuit) when applying the formulas. Good luck!

1. Key Concepts and Formulas to Recall

Before attempting any problem, ensure you are thoroughly familiar with these fundamental relationships:

• Electromotive Force (EMF, $mathcal{E}$): The maximum potential difference a cell can provide when no current is drawn from it (open circuit).


• Terminal Voltage (V): The actual potential difference across the terminals of a cell when current is flowing.
  
  
  
  • Important: $V le mathcal{E}$. Equality holds only in an open circuit.
  
  
  
  


• Internal Resistance (r): The resistance offered by the electrolyte and electrodes within the cell.


• Ohm's Law for a complete circuit: $I = frac{mathcal{E}}{R + r}$, where $R$ is the external resistance.


• Relation between EMF and Terminal Voltage: $V = mathcal{E} - Ir$


• Power delivered by the cell to the external circuit: $P_{ext} = IV = I^2 R = frac{V^2}{R}$


• Power dissipated within the cell (due to internal resistance): $P_{int} = I^2 r$


• Total power produced by the cell: $P_{total} = Imathcal{E} = P_{ext} + P_{int}$


• Condition for Maximum Power Transfer: Maximum power is delivered to the external load when $R = r$. In this case, $P_{max} = frac{mathcal{E}^2}{4r}$. (JEE Specific)

2. Step-by-Step Problem-Solving Methodology

1. Analyze the Circuit and Identify Variables:
   
   
   
   • Draw a clear circuit diagram if not provided.
   
   
   • Identify all given quantities: EMF ($mathcal{E}$), internal resistance (r), external resistance (R), current (I), terminal voltage (V).
   
   
   • Clearly mark what needs to be calculated.
   
   
   • Note the direction of current flow.
   
   
   
   


2. Apply Ohm's Law for the Full Circuit:
   
   
   
   • The most common starting point is $I = frac{mathcal{E}}{R_{eq} + r_{eq}}$.
   
   
   • If there are multiple external resistors, find their equivalent resistance ($R_{eq}$).
   
   
   • If there are multiple cells, find their equivalent EMF and internal resistance ($r_{eq}$) based on series or parallel connection. (JEE Specific: Combinations of cells)
   
   
   
   


3. Calculate Terminal Voltage:
   
   
   
   • Once current $I$ is known, use $V = mathcal{E} - Ir$.
   
   
   • Alternatively, the terminal voltage is also the voltage drop across the external resistance, $V = IR_{eq}$. This equality is a good cross-check.
   
   
   
   


4. Power Calculations (If required):
   
   
   
   • Use the appropriate power formula ($P_{ext} = I^2R$, $P_{int} = I^2r$, $P_{total} = Imathcal{E}$).
   
   
   • For maximum power transfer, verify if $R=r$.
   
   
   
   


5. Consider Special Cases:
   
   
   
   • Open Circuit: $I=0$, so $V = mathcal{E}$. The voltmeter reading directly gives the EMF.
   
   
   • Short Circuit: $R=0$, so $I_{sc} = frac{mathcal{E}}{r}$. This scenario is often used to find 'r'.
   
   
   
   

3. Common Pitfalls & JEE Specific Tips

• Confusing EMF and Terminal Voltage: Always remember $V = mathcal{E}$ only when no current flows ($I=0$). Otherwise, $V < mathcal{E}$.


• Sign Conventions in Kirchhoff's Laws: When applying Kirchhoff's Voltage Law (KVL) to circuits with cells, be meticulous with sign conventions for EMF and voltage drops across resistors (internal and external).


• Cells in Series/Parallel: Understand how equivalent EMF and internal resistance are calculated for different configurations. For series aiding, $mathcal{E}_{eq} = sum mathcal{E}_i$, $r_{eq} = sum r_i$. For parallel (identical cells), $mathcal{E}_{eq} = mathcal{E}$, $r_{eq} = r/n$. (JEE Specific)


• Maximum Power Transfer Theorem: Recognize problems asking for conditions or value of external resistance for maximum power output. This is a common JEE question.

JEE Focus Areas: Cell, EMF and Internal Resistance

This section is critical for understanding the behavior of real-world cells and batteries in circuits. JEE Main frequently tests your conceptual clarity and problem-solving skills based on these topics. Mastering the relationships between EMF, terminal voltage, current, and internal resistance is essential.

1. Key Concepts & Definitions

• Electromotive Force (EMF, E):
  
  
  
  • The maximum potential difference that a cell can provide across its terminals when no current is drawn from it (i.e., in an open circuit).
  
  
  • It represents the energy supplied per unit charge by the non-electrical forces inside the cell.
  
  
  • JEE Tip: EMF is a source characteristic and is *not* a force; it's a potential difference measured in Volts.
  
  
  
  


• Internal Resistance (r):
  
  
  
  • The resistance offered by the electrolyte and electrodes within the cell to the flow of current.
  
  
  • It leads to a drop in potential difference across the cell terminals when current flows.
  
  
  • Units: Ohms (Ω).
  
  
  
  


• Terminal Voltage (V) or Potential Difference:
  
  
  
  • The actual potential difference across the cell terminals when current is being drawn from it.
  
  
  • It is always less than the EMF when the cell is discharging (V < E).
  
  
  
  

2. Fundamental Relationships (Discharging Cell)

For a cell with EMF (E) and internal resistance (r) connected to an external resistance (R) through which current (I) flows:

• Relationship between V, E, I, and r:
  

  $$V = E - Ir$$

  
  
  
  
  • Here, 'Ir' is the voltage drop across the internal resistance.
  
  
  • This is one of the most frequently used formulas in JEE.
  
  
  
  


• Current in the circuit:
  

  $$I = frac{E}{R + r}$$

  
  
  
  
  • This combines the cell and external resistance into a simple series circuit.
  
  
  
  


• Power delivered to the external circuit:
  

  $$P_{external} = I^2 R = VI = left(frac{E}{R+r}
  ight)^2 R$$

  
  


• Power dissipated internally:
  

  $$P_{internal} = I^2 r$$

  
  


• Total power generated by the cell:
  

  $$P_{total} = EI = P_{external} + P_{internal}$$

  
  

3. Important Cases & Scenarios

• Open Circuit:
  
  
  
  • No current flows (I = 0).
  
  
  • Terminal voltage equals EMF (V = E). This is how EMF is practically measured.
  
  
  
  


• Short Circuit:
  
  
  
  • External resistance R = 0 (terminals directly connected).
  
  
  • Maximum current flows: $$I_{sc} = frac{E}{r}$$
  
  
  • Terminal voltage V = 0.
  
  
  
  


• Charging of a Cell:
  
  
  
  • When an external source pushes current (I) into the cell (opposite to its natural discharge direction).
  
  
  • Terminal voltage becomes greater than EMF: $$V = E + Ir$$
  
  
  • JEE Focus: Be careful with the sign of 'Ir' depending on whether the cell is charging or discharging.
  
  
  
  


• Maximum Power Transfer Theorem:
  
  
  
  • The power delivered to the external resistance (R) is maximum when the external resistance is equal to the internal resistance of the cell (R = r).
  
  
  • Maximum power: $$P_{max} = frac{E^2}{4r}$$
  
  
  • This is a very common JEE problem type.
  
  
  
  

4. Graphical Analysis (V vs I plot)

The equation V = E - Ir represents a straight line of the form y = mx + c.

• Plotting terminal voltage (V) on the y-axis and current (I) on the x-axis gives a straight line.


• Y-intercept: E (EMF, when I=0).


• Slope: -r (Negative of internal resistance).


• X-intercept: $$I_{sc} = E/r$$ (Short circuit current, when V=0).


• JEE Tip: Be prepared to interpret such graphs to find E and r, or to predict V for a given I.

Understanding these relationships and scenarios thoroughly will enable you to solve a wide range of problems related to cells and batteries in JEE Main.