A cell of e.m.f. E is connected with an external resistance R , then p.d. across cell is V . The internal resistance of cell will be

Options

A. `((E-V)R)/(E)`
B. `((E-V)R)/(V)`
C. `((V-E)R)/(V)`
D. `((V-E)R)/(E)`

Answer

To find the internal resistance of the cell, we can use the relationship between the electromotive force (e.m.f.) of the cell, the potential difference (p.d.) across the cell, the external resistance, and the internal resistance. Here’s a step-by-step solution:

### Step 1: Understand the relationship between e.m.f., p.d., and resistances
The e.m.f. (E) of the cell is the maximum potential difference when no current is flowing. When the cell is connected to an external resistance (R), the potential difference (V) across the terminals of the cell decreases due to the internal resistance (r) of the cell.

### Step 2: Write the equation for the circuit
According to Kirchhoff's voltage law, the sum of the potential differences in a closed circuit is equal to zero. Therefore, we can write:
\[ E = V + I \cdot r \]
Where:
- \( E \) = e.m.f. of the cell
- \( V \) = potential difference across the cell
- \( I \) = current flowing through the circuit
- \( r \) = internal resistance of the cell

### Step 3: Express the current (I)
The current \( I \) can be expressed using Ohm's law:
\[ I = \frac{V}{R} \]
Where \( R \) is the external resistance.

### Step 4: Substitute the expression for I into the equation
Substituting \( I \) into the equation from Step 2 gives:
\[ E = V + \left(\frac{V}{R}\right) \cdot r \]

### Step 5: Rearrange the equation to solve for internal resistance (r)
Rearranging the equation:
\[ E - V = \frac{V}{R} \cdot r \]
Now, multiply both sides by \( R \):
\[ R(E - V) = V \cdot r \]
Now, divide both sides by \( V \):
\[ r = \frac{R(E - V)}{V} \]

### Final Answer
The internal resistance \( r \) of the cell is given by:
\[ r = \frac{R(E - V)}{V} \]

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