A battery has an emf E and internal resistance r. A variable resistance R is connected across the terminals of the battery. Find the value of R such that (a) the current in the circuit id maximum (b) the potential difference across the terminals is maximum.

Explanation

To solve the problem, we need to analyze the circuit consisting of a battery with an electromotive force (emf) \( E \) and internal resistance \( r \), connected to a variable resistance \( R \). We will find the value of \( R \) for two scenarios: (a) when the current in the circuit is maximum, and (b) when the potential difference across the terminals is maximum.

### Step-by-Step Solution:

#### Part (a): Finding \( R \) for Maximum Current

1. **Understanding the Circuit**:
The total resistance in the circuit is the sum of the internal resistance \( r \) and the variable resistance \( R \). The current \( I \) flowing through the circuit can be expressed using Ohm's law:
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