Question

In the experiment of potentiometer wire \(\mathrm{AB}\) is \(100 \mathrm{~cm}\) long shown in figure When \(\mathrm{AC}=40 \mathrm{~cm}\), no deflection occurs in the galvanometer. What is the value of \(R\) ? (A) \(15 \Omega\) (B) \(18 \Omega\) (C) \(12 \Omega\) (D) \(14 \Omega\)

Short answer

The value of R is \(15\Omega\), which corresponds to option (A).

Step-by-step solution

Understanding the potentiometer experiment

In the potentiometer experiment, a null-point (no deflection) in the galvanometer indicates that the potential difference across the known length of the wire (AC in this case) and the unknown resistance (R) is equal. Thus, we can set up an equation: \( \frac{V_{AC}}{AC} = \frac{V_{BR}}{BR} \) Where V_AC is the potential difference across the length AC, V_BR is the potential difference across length BR, and BR is the length of the wire between points B and R.

Determine the lengths of AC and BR

Since AC is given as 40 cm and the total length of the wire, AB, is 100 cm, we can find the length of BR as: \( BR = AB - AC \) \( BR = 100cm - 40cm = 60cm \)

Calculate the potential difference across AC and BR

As there is no deflection in the galvanometer, the potential difference across AC and R, V_AC, is equal to the potential difference across BR, V_BR. Therefore, we can write the initial equation as: \( \frac{V}{40} = \frac{V}{60} \) where V is the potential difference across both AC and BR. Solving for V, we get: \( V = \frac{2}{3} V \)

Find the value of R using Ohm's law

Using Ohm's law, we can write the equation for the potential difference across the unknown resistance R as: \( V_{BR} = I \times R \) where I is the current flowing through the circuit. Substituting the expression for V_BR from step 3, we get: \( \frac{2}{3}V = I \times R \) Since the potential difference across both AC and BR is equal, we can write another equation for the potential difference using Ohm's law for length AC: \( V_{AC} = I \times 10 \) Where the resistance of AC is \( 10\Omega \) (given in the problem).

Solve for R

Now, we are given that the potential difference between A and C is zero. This means that the potential difference is the same for both lengths of the wire AC and BR. Therefore, \( V_{AC}= V_{BR} \) . Hence, we can write our equation as: \( I \times 10 = \frac{2}{3}V = I \times R \) Now, divide both sides by I: \( 10 = \frac{2}{3}R \) Multiplying both sides by 3 and dividing by 2, we get: \( R = \frac{3}{2} \times 10 = 15\Omega \) Hence, the value of R is 15Ω, which corresponds to option (A).