Question

To determine the resistance $R$ of a wire, a circuit is designed below. The V-I characteristic curve for this circuit is plotted for the voltmeter and the ammeter readings as shown in the figure. The value of $R$ is $\dots \dots \dots \Omega$.

Answer 1

The equivalent resistance $R_\text{eq}$ of two resistors in parallel is given by:
$R_\text{eq} = \frac{10^4 R}{10^4 + R}.$
Given: $E = 4 \, \text{V}, \quad I = 2 \, \text{mA}.$
From Ohm's Law: $I = \frac{E}{R_\text{eq}}.$
Substitute $R_\text{eq}$ into the equation:
$2 \times 10^{-3} = \frac{4}{\frac{10^4 R}{10^4 + R}}.$
Simplify: $2 \times 10^{-3} = \frac{4(10^4 + R)}{10^4 R}.$
Multiply through by $10^4 R$: $2 \times 10^4 R = 4(10^4 + R).$
Distribute and simplify: $20R = 40000 + 4R.$
Rearranging terms: $16R = 40000.$
Solve for $R$: $R = \frac{40000}{16} = 2500 \, \Omega.$
Thus, the resistance $R$ is: $\boxed{2500 \, \Omega}.$

Explanation: The equivalent resistance for two parallel resistors is determined by the formula $R_\text{eq} = \frac{10^4 R}{10^4 + R}$. By using the provided voltage and current values, Ohm's Law was applied to derive $R$. The algebraic simplifications lead to $R = 2500 \, \Omega$.

Answer 2

The equivalent resistance $R_\text{eq}$ of two resistors in parallel is given by:
$R_\text{eq} = \frac{10^4 R}{10^4 + R}.$
Given: $E = 4 \, \text{V}, \quad I = 2 \, \text{mA}.$
From Ohm's Law: $I = \frac{E}{R_\text{eq}}.$
Substitute $R_\text{eq}$ into the equation:
$2 \times 10^{-3} = \frac{4}{\frac{10^4 R}{10^4 + R}}.$
Simplify: $2 \times 10^{-3} = \frac{4(10^4 + R)}{10^4 R}.$
Multiply through by $10^4 R$: $2 \times 10^4 R = 4(10^4 + R).$
Distribute and simplify: $20R = 40000 + 4R.$
Rearranging terms: $16R = 40000.$
Solve for $R$: $R = \frac{40000}{16} = 2500 \, \Omega.$
Thus, the resistance $R$ is: $\boxed{2500 \, \Omega}.$

Explanation: The equivalent resistance for two parallel resistors is determined by the formula $R_\text{eq} = \frac{10^4 R}{10^4 + R}$. By using the provided voltage and current values, Ohm's Law was applied to derive $R$. The algebraic simplifications lead to $R = 2500 \, \Omega$.