Problem 99 - A Complete Resource Book in Physics for JEE Main

Question

An ammeter is obtained by shunting a \(30 \Omega\) galvanometer with a \(30 \Omega\) resistance. What additional shunt should be connected across it to double the range? (A) \(15 \Omega\) (B) \(10 \Omega\) (C) \(5 \Omega\) (D) None of these

Step-by-Step Solution

Answer

The additional shunt resistance needed to double the range of the ammeter is \(10 \Omega\), which is option (B).

Step 1: Understand the initial setup

We are given that the ammeter is formed by shunting a \(30 \Omega\) galvanometer with a \(30 \Omega\) resistance. In this case, the galvanometer and the shunt resistance are connected in parallel, and the total resistance RT of this ammeter can be calculated using the parallel resistance formula: \[ \frac{1}{R_T} = \frac{1}{R_G} + \frac{1}{R_S} \] Where, \(R_T\) = total resistance of the ammeter \(R_G\) = resistance of the galvanometer (\(30 \Omega\)) \(R_S\) = initial shunt resistance (\(30 \Omega\))

Step 2: Calculate the total resistance of the ammeter

Now let's calculate the total resistance of the ammeter using the formula from Step 1: \[ \frac{1}{R_T} = \frac{1}{30 \Omega} + \frac{1}{30 \Omega} \] \[ R_T = \frac{1}{\frac{1}{30} + \frac{1}{30}} = \frac{1}{\frac{2}{30}} = 15 \Omega \] So the total resistance of the ammeter with the initial setup is \(15 \Omega\).

Step 3: Write the formula for doubling the range

In order to double the range of the ammeter, we need to find a new shunt resistance, say \(R_{S2}\), that when connected in parallel with the initial resistance \(R_S\), results in half of the initial total resistance (\(\frac{15}{2}\) \(\Omega\)). This can be written using the parallel resistance formula: \[ \frac{1}{R_{S2} + R_S} = \frac{1}{R_G} - \frac{1}{R_T'} \] Where, \(R_{S2}\) = new additional shunt resistance \(R_T'\) = new total resistance after doubling the range

Step 4: Calculate the new total resistance

Since we want to double the range of the ammeter, the new total resistance should be half of the initial resistance: \[ R_T' = \frac{15}{2} \Omega = 7.5 \Omega \]

Step 5: Find the value of the additional shunt resistance

Now, we can use the formula from Step 3, substituting the known values to find the additional shunt resistance needed to double the range: \[ \frac{1}{R_{S2} + 30 \Omega} = \frac{1}{30 \Omega} - \frac{1}{7.5 \Omega} \] Now we can solve for \(R_{S2}\): \[ R_{S2} = \frac{1}{\frac{1}{30}-\frac{1}{7.5}} - 30 = 10 \Omega \] So the additional shunt resistance needed to double the range of the ammeter is \(10 \Omega\), which is option (B).