Question Video: Recalling the Effect of Combining a Shunt Resistor with a Galvanometer Physics

Video Transcript

Which of the following is the most correct description of how the range of values of currents that a galvanometer can produce is extended when it is converted into an ammeter using a shunt resistor? (A) A shunt resistor with a resistance much smaller than that of the galvanometer is connected in parallel with the galvanometer. (B) A shunt resistor with a resistance much greater than that of the galvanometer is connected in parallel with the galvanometer. (C) A shunt resistor with a resistance much smaller than that of the galvanometer is connected in series with the galvanometer. (D) A shunt resistor with a resistance much greater than that of the galvanometer is connected in series with the galvanometer. (E) A shunt resistor with a resistance equal to that of the galvanometer is connected in parallel with the galvanometer.

In this question, we want to work out where we can place the shunt resistor in the circuit in order to extend the range of currents that the galvanometer can produce. Let’s begin by connecting a shunt resistor in series with the galvanometer, as is described in options (C) and (D).

Recall that when we have two resistors connected in series, the total resistance is given by the sum of the individual resistances. So, by labeling the resistance of the galvanometer as 𝑅 sub 𝐺 and the resistance of the shunt resistor as 𝑅 sub 𝑆, we have that the total resistance 𝑅 sub total is equal to 𝑅 sub 𝐺 plus 𝑅 sub 𝑆.

Also, recall that Ohm’s law can be written as 𝑉 equals 𝐼 times 𝑅, where 𝑉 is the potential difference, 𝐼 is the current in the circuit, and 𝑅 is the resistance. We can rearrange this to get an expression for the current in the circuit. To do this, we just need to divide both sides of the equation by the resistance 𝑅. This gives us the expression 𝐼 is equal to 𝑉 over 𝑅. Applying Ohm’s law across this circuit, we get 𝐼 is equal to 𝑉 over 𝑅 sub total.

If we pick a shunt resistor with a resistance much smaller than that of the galvanometer, that is, when we pick a shunt resistor such that 𝑅 sub 𝑆 is much less than 𝑅 sub 𝐺, then 𝑅 sub total is equal to 𝑅 sub 𝐺. In this case, the shunt resistor has pretty much no effect on the resistance of the circuit, meaning it does not change the range of current values that the galvanometer can produce. So, option (C) is incorrect.

If we pick a shunt resistor with a resistance much larger than that of the galvanometer, that is, if we pick a shunt resistor such that 𝑅 sub 𝑆 is much greater than 𝑅 sub 𝐺, then 𝑅 sub total is equal to 𝑅 sub 𝑆. In this case, the denominator in our expression for the current will be much larger than before. This means that adding this resistor in series will make the current much smaller. In other words, this decreases the range of current values that the galvanometer can produce. So, option (D) is incorrect.

So now, let’s try connecting the shunt resistor in parallel with the galvanometer, as is described in options (A), (B), and (E). Recall that the current splits across parallel paths. We can apply Ohm’s law to each path separately to find out how much current is in each path. We’ll call the current through the galvanometer 𝐼 sub 𝐺 and the current through the shunt resistor 𝐼 sub 𝑆. So, the currents through the galvanometer and shunt resistor are 𝐼 sub 𝐺 is equal to 𝑉 sub 𝐺 over 𝑅 sub 𝐺 and 𝐼 sub 𝑆 is equal to 𝑉 sub 𝑆 over 𝑅 sub 𝑆, respectively.

We know that both paths have the same potential difference, so this means 𝑉 sub 𝐺 equals 𝑉 sub 𝑆, which is equal to 𝑉, the potential difference provided by the cells. So, the current through the galvanometer is equal to 𝑉 over 𝑅 sub 𝐺, and the current through the shunt resistor is equal to 𝑉 over 𝑅 sub 𝑆.

Let’s consider what happens when we choose a shunt resistor with a resistance much smaller than that of the galvanometer, that is, when we pick a shunt resistor such that 𝑅 sub 𝑆 is much less than 𝑅 sub 𝐺. Since 𝐼 sub 𝐺 and 𝐼 sub 𝑆 have the same values in the numerator, this means that if 𝑅 sub 𝑆 is much less than 𝑅 sub 𝐺, then 𝐼 sub 𝑆 is much greater than 𝐼 sub 𝐺. In other words, most of the current goes through the path containing the shunt resistor.

Meanwhile, there is a small current through the galvanometer. The current that passes through the galvanometer is a constant proportion of the current in the circuit. This means that the deflection of the galvanometer’s needle will be proportional to the current in the circuit. So, if a large current is passed through the circuit, only a small proportion of it will pass through the galvanometer. This means that a much larger current can pass through the circuit before the galvanometer reaches full-scale deflection.

Hence, the combination of the galvanometer and shunt resistor in parallel may be used to extend the range of values of currents that a galvanometer can produce. Therefore, option (A) is the most correct description. A shunt resistor with a resistance much smaller than that of the galvanometer is connected in parallel with the galvanometer.