Question

The magnetic flux through a coil perpendicular to its plane and directed into paper is varying according to the relation $\phi = \left( {2{t^2} + 4t + 6} \right){\text{m Wb}}$. The e.m.f induced in the loop at $t = 4s$ is,
A) $1.2{\text{V}}$.
B) $2.4{\text{V}}$.
C) $0.02{\text{V}}$.
D) $1.2{\text{V}}$.

Answer 1

Hint:The formula of induced e.m.f given by Faraday can be used to calculate the correct answer to this problem. Faraday's law says that the rate of change of magnetic field is given by the induced e.m.f. The induced e.m.f is such that there is oppose of the magnetic field by the coil in which the e.m.f is induced.

Formula:The formula for induced e.m.f is given by, $e = - \dfrac{{d\phi }}{{dt}}$ where $\phi $ is magnetic field and $t$ is time taken. The negative sign shows that the induced e.m.f will oppose the magnetic field.

Step by step solution:
Step 1.
As it is given that magnetic field $\phi = \left( {2{t^2} + 4t + 6} \right){\text{m Wb}}$ and time is $t = 4s$. The magnetic flux is given in milli weber.
Step 2.
First we need to differentiate the equation with respect to time.
Applying,
$e = \dfrac{{d\phi }}{{dt}}$
Put the value of magnetic field $\phi $ in the above relation,
$
  e = \dfrac{{d\phi }}{{dt}} \\
  e = \dfrac{{d\left( {2{t^2} + 4t + 6} \right) \times {{10}^{ - 3}}}}{{dt}} \\
  e = \left( {4t + 4} \right) \times {10^{ - 3}} \\
 $………eq (1)
Step 3.
As induced e.m.f has to be calculated at $t = 4s$,
Put $t = 4s$ in the equation (1),
\[
  e = \left( {4t + 4} \right) \times {10^{ - 3}} \\
  e = \left( {4 \cdot 4 + 4} \right) \times {10^{ - 3}} \\
  e = 20 \times {10^{ - 3}} \\
  e = 0.02{\text{V}} \\
 \]
So, the induced e.m.f is \[e = 0.02{\text{V}}\]

therefore the correct answer for this problem is option C.

Additional information: Faraday’s law states that induced e.m.f is equal to the rate of change of magnetic field with respect to time and the induced e.m.f opposes the magnetic field.

Note: While calculating we remove the negative sign as we need to calculate the magnitude of the induced e.m.f and the direction of induced e.m.f is such that it opposes the magnetic field. Students should remember Faraday’s law as this law defines the working of most modern day appliances.