Question

A coil having $n$ turns and resistance $R$ is connected with a galvanometer of resistance $4R$. This combination is moved in time $t$ seconds from a magnetic flux ${{W}_{1}}$ to ${{W}_{2}}$. The induced current in the circuit is
A. $\dfrac{{{W}_{2}}-{{W}_{1}}}{5Rnt}$
B. $\dfrac{n({{W}_{2}}-{{W}_{1}})}{5Rt}$
C. $\dfrac{{{W}_{2}}-{{W}_{1}}}{Rnt}$
D. $\dfrac{n({{W}_{2}}-{{W}_{1}})}{Rnt}$

Answer 1

Use the Faraday’s law of electro-magnetism for induced emf inside a coil due to a changing magnetic flux through the coil. Then use the Ohm’s law to find an expression of current in terms of rate of change in magnetic flux.

Formula used:
$E=n\dfrac{d\phi }{dt}$
where E is induced emf in a coil with n turn when the rate of change in magnetic flux through the coil is $\dfrac{d\phi }{dt}$.
$E=iR$
where E is emf across a resistance R when current i is flowing through it.

Complete step by step answer:
When the magnetic flux through a coil changes with time, an emf is induced in the coil during that time. According to the Faraday’s law of electro-magnetism, the magnitude of the induced emf in a coil of n turns due to changing magnetic flux through the coil is given as,
$E=n\dfrac{d\phi }{dt}$
If the rate of change in the magnetic flux is uniform or constant that the induced emf is given as $E=n\dfrac{\Delta \phi }{\Delta t}$ …. (i)
where $\Delta \phi$ is the change in the magnetic flux in a time interval of $\Delta t$
And from Ohm’s law we know that $E=iR$ …. (ii).
Now, equate (i) and (ii).
With this we get, $n\dfrac{\Delta \phi }{\Delta t}=iR$.
Then,
$i=\dfrac{n}{R}\dfrac{\Delta \phi }{\Delta t}$.
In the question, it is given that the coil contains a resistance R and galvanometer of resistance 4R. Let us assume that the two are connected in series. Then the total resistance of the coil is equal to 5R.
Also, $\Delta \phi ={{W}_{2}}-{{W}_{1}}$ and $\Delta t=t$
With the substitutions we get that $i=\dfrac{n({{W}_{2}}-{{W}_{1}})}{5Rt}$
Therefore, the current in the coil is equal to $\dfrac{n({{W}_{2}}-{{W}_{1}})}{5Rt}$

Hence, the correct option is B.

Note: It is necessary that we assume that the galvanometer and the coil are connected in series. Otherwise, the resistance of the system will not be known.You may also consider the galvanometer to be connected in parallel. However, a galvanometer is mostly used in series to detect current.