NCERT Solutions

We need to calculate the resistor's resistance and the battery's terminal voltage from the provided emf, internal resistance, and current.

Concept: Electromotive Force

The relationship between Electromotive Force ($\mathcal{E}$), Internal Resistance ($r$), and external resistance ($R$) is given by Ohm's Law for a complete circuit:

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

The Terminal Voltage ($V$) is the potential difference across the battery terminals when current flows:

$$V = \mathcal{E} - Ir$$

Given:

• Electromotive force, $\mathcal{E} = 10\,\text{V}$
• Internal resistance, $r = 3\,\Omega$
• Current, $I = 0.5\,\text{A}$

Solving:

1. Calculate the resistance of the resistor ($R$):

   Using the circuit equation:

$$\begin{aligned} I &= \frac{\mathcal{E}}{R + r} \\ 0.5\,\text{A} &= \frac{10\,\text{V}}{R + 3\,\Omega} \\ R + 3\,\Omega &= \frac{10\,\text{V}}{0.5\,\text{A}} \\ R + 3\,\Omega &= 20\,\Omega \\ R &= 20\,\Omega - 3\,\Omega \\ R &= 17\,\Omega \end{aligned}$$

2. Calculate the terminal voltage of the battery ($V$):

   Using the terminal voltage formula:

$$\begin{aligned} V &= \mathcal{E} - Ir \\ V &= 10\,\text{V} - (0.5\,\text{A} \cdot 3\,\Omega) \\ V &= 10\,\text{V} - 1.5\,\text{V} \\ V &= 8.5\,\text{V} \end{aligned}$$

Alternatively, using $V = IR$:

$$\begin{aligned} V &= 0.5\,\text{A} \cdot 17\,\Omega \\ V &= 8.5\,\text{V} \end{aligned}$$

Result:

The resistance of the resistor is $17\,\Omega$ and the terminal voltage of the battery is $8.5\,\text{V}$.

$$R = 17\,\Omega, \quad V = 8.5\,\text{V}$$