Question

A rectangular coil of 20 turns and an area of cross-section 25 sq. cm has a resistance of 100 Ohm. If a magnetic field which is perpendicular to the plane of the coil changes at a rate of 1000 tesla per sec, the current in the coil is:
A. 1 amp
B. 50 amp
C. 0.5 amp
D. 5.0 amp

Answer 1

Hint: The current is the emf by the resistance. The emf is the product of the number of turns of a coil, the area of the coil and the rate of change of the magnetic field. We will combine these equations and will substitute the given values to find the value of the current in the coil.
Formula used:
\[\begin{align}
  & i=\dfrac{\varepsilon }{R} \\
 & \varepsilon =NA\dfrac{dB}{dt} \\
\end{align}\]

Complete answer:
From the given information, we have the data as follows.
A rectangular coil of 20 turns and area of cross section 25 sq.cm has a resistance of 100 Ohm. A magnetic field that is perpendicular to the plane of the coil changes at a rate of 1000 tesla per sec.
\[\begin{align}
  & N=20 \\
 & A=25\,c{{m}^{2}}=25\times {{10}^{-4}}{{m}^{2}} \\
 & R=100\,\Omega \\
 & \dfrac{dB}{dt}=1000\,T{{s}^{-1}} \\
\end{align}\]
The formula that relates the parameters current, resistance and the emf is given as follows.
\[i=\dfrac{\varepsilon }{R}\]
Where i is current, R is the resistance and \[\varepsilon \] is the emf.
The emf in terms of the number of turns in a coil, the change in the magnetic field and the change in the magnetic field is given as follows.
\[\varepsilon =NA\dfrac{dB}{dt}\]
Where N is the number of turns in a coil, A is the area of the coil and \[\dfrac{dB}{dt}\] is the change in the magnetic field.
Substitute the equation of the emf in the equation of the current.
\[i=\dfrac{NA\dfrac{dB}{dt}}{R}\]
Now, substitute the given values of the number of turns in a coil, area of the rectangular coil, the change in the magnetic field and the resistance of the coil in the above equation.
\[i=\dfrac{20\times 25\times {{10}^{-4}}\times 1000}{100}\]
Continue further computation.
\[\begin{align}
  & i=2\times 25\times 0.01 \\
 & \therefore i=0.5\,A \\
\end{align}\]
\[\therefore \] The current in the rectangular coil is 0.5 A.

Thus, option (C) is correct.

Note:
The units of the parameters should be taken care of. As in this case, we have changed the unit of the area from cm to meters. The units of the remaining parameters are in the SI unit. The value of the rate of change of the magnetic field can be given separately as the value of the magnetic field and the value of time.