Question

The area of a 120-turn coil oriented with it's plane perpendicular to a 0.20-T magnetic field is 0.050 m². Find the average induced emf in this coil if the magnetic field reverses its direction in 0.34s.

Answer 1

Concepts and reason

The main concepts that are required to solve this problem are the magnetic flux, time, magnetic field, area and induced emf.

Initially, write the equation for the magnetic flux and the induced emf due to the change in magnetic flux in the coil and use these equations and calculate the induced emf in the given coil if the magnetic field reverses its direction.

Fundamentals

The equation for the magnetic flux is,

$\phi = BA$

Here, B is the magnetic field and A is the area through which the magnetic field lines are passing.

The equation for the induced emf in the coil due to the change in magnetic flux is,

$\varepsilon = - N\frac{{\Delta \phi }}{{\Delta t}}$

Here, N is the number of coils, $\Delta \phi$ is the change in magnetic flux and $\Delta t$ is the time interval and the minus sign indicates the Lenz’s law.

The equation for the change in magnetic flux is,

$\Delta \phi = {\phi _f} - {\phi _i}$

Here, ${\phi _f}$ is the final magnetic flux and ${\phi _i}$ is the initial magnetic flux.

The equation for the induced emf in the coil is,

$\varepsilon = - N\frac{{\Delta \phi }}{{\Delta t}}$ …… (1)

The equation for the change in magnetic flux is,

$\Delta \phi = {\phi _f} - {\phi _i}$ …… (2)

The equation for the initial magnetic flux is,

${\phi _i} = BA$

Here, B is the magnetic field and A is the area.

The equation for the final magnetic flux if the magnetic field reverses its direction is,

${\phi _f} = - BA$

Here, the minus indicates the direction which was reversed.

Substitute $BA$ for ${\phi _i}$ , and $- BA$ for ${\phi _f}$ in above equation (2).

$\Delta \phi = - BA - BA$

$= - 2BA$ …… (3)

Substitute the equation (3) in above equation (1).

$\begin{array}{c}\\\varepsilon = - N\frac{{\left( { - 2BA} \right)}}{{\Delta t}}\\\\ = \frac{{2NBA}}{{\Delta t}}\\\end{array}$

The equation for the average induced emf that derived in the step 1 is,

$\varepsilon = \frac{{2NBA}}{{\Delta t}}$

Here, N is the number of coils, B is the magnetic field, A is the area and $\Delta t$ is the time interval.

Substitute 120 for N, 0.20 T for B, $0.050{\rm{ }}{{\rm{m}}^2}$ for A, and $0.34{\rm{ s}}$ for $\Delta t$ in above equation.

$\begin{array}{c}\\\varepsilon = \frac{{2\left( {120} \right)\left( {0.20{\rm{ T}}} \right)\left( {0.050{\rm{ }}{{\rm{m}}^2}} \right)}}{{\left( {0.34{\rm{ s}}} \right)}}\\\\ = 7.0{\rm{ V}}\\\end{array}$

Ans:

The average induced emf in the coil if the magnetic field reverses its direction is $7.0{\rm{ V}}$ .