Question

A galvanometer is used to measure small currents. A certain galvanometer has a resistance of $500\Omega$ and gives a full-scale deflection for a current of $200\mu A$. This meter is connected as shown in the figure to make a multirange current meter.
Connection to the circuit is made at the terminals shown. The currents in the external circuit needed to give full-scale deflections when $X$ is connected to $A,B$ and $C$, in turn, are shown in the table.

$X$ connected to	Current in the external circuit $\left(mA\right)$
$A$	$1$
$B$	$10$
$C$	$100$

Find out the of $R_3$ is

• A. $2.25\Omega$
• B. $0.25\Omega$
• C. $1.25\Omega$
• D. $3.25\Omega$

Answer 1

The correct option is C $1.25\Omega$

Given: A certain galvanometer has a resistance of $500\Omega$ and gives a full-scale deflection for a current of $200\mu A$. This meter is connected as shown in the figure to make a multi-range current meter.

Connection to the circuit is made at the terminals shown. The currents in the external circuit needed to give full-scale deflections when X is connected to A,B and C, in turn, are shown in the table.

To find $R_3$

Solution:

The given circuit can be re-written as shown in fig(i),

Now applying Kirchhoff's Law, we get

$500\times 200\mu A=(R_1+R_2+R_3)i_1⟹500\times 200\times {10}^{-6}A=(R_1+R_2+R_3)[1ma-200\mu A]⟹R_1+R_2+R_3=\frac{10\times {10}^4\times {10}^{-6}}{{10}^{-6}[1000-200]}⟹R_1+R_2+R_3=\mathrm{125....}\left(i\right)$

Now, according to the given criteria, when X is connected to B the circuit becomes as shown in fig(ii), and we get

$(500+R_1)200\mu A=(R_2+R_3)[10mA-200\mu A]⟹(500+R_1)200\times {10}^{-6}=(R_2+R_3){10}^{-6}[10000-200]⟹500+R_1=49(R_2+R_3)....\left(ii\right)$

Again when X is connected to C the circuit becomes as shown in fig(iii), and we get

$(500+R_1+R_2)200\mu A=\left(R_3\right)[100mA-200\mu A]⟹(500+R_1+R_2)200\times {10}^{-6}=\left(R_3\right){10}^{-6}[100000-200]⟹500+R_1+R_2=499\left(R_3\right)....\left(iii\right)$

From (i) and (iii),

$500+125-R_3=499R_3⟹500R_3=625⟹R_3=1.25\Omega$

is the required value.