A closed flat loop conductor with radius $2.0 \mathrm {~m}$ is located in a changing uniform magnetic field. If the emf induced in the loop is $2.0 \mathrm {~V}$ what is the rate at which the magnetic field strength is changing if the magnetic field is oriented perpendicular to the plane in which the loop lies?
A) 0.16 T/s
B) 1.0 T/s
C) 0.080 T/s
D) 2.0 T/s

Question

Quizplus · Accepted Answer

The emf induced in a closed loop conductor is given by the equation:$$\varepsilon = - \frac{d\Phi_B}{dt}$$where $\varepsilon$ is the emf, $\Phi_B$ is the magnetic flux through the loop, and $t$ is time.In this case, the magnetic field is perpendicular to the plane of the loop, so the magnetic flux is given by:$$\Phi_B = BA$$where $B$ is the magnetic field strength and $A$ is the area of the loop.Substituting this into the first equation, we get:$$\varepsilon = - \frac{d(BA)}{dt}$$$$\varepsilon = -B\frac{dA}{dt} - A\frac{dB}{dt}$$Since the area of the loop is constant, we have:$$\varepsilon = -A\frac{dB}{dt}$$Solving for $\frac{dB}{dt}$, we get:$$\frac{dB}{dt} = -\frac{\varepsilon}{A}$$Substituting the given values, we get:$$\frac{dB}{dt} = -\frac{2.0 \mathrm{~V}}{\pi (2.0 \mathrm{~m})^2} = -0.16 \mathrm{~T/s}$$Therefore, the rate at which the magnetic field strength is changing is 0.16 T/s.

A Closed Flat Loop Conductor with Radius 2.0 m2.0 \mathrm {~m}2.0 m Is Located in a Changing Uniform Magnetic Field

A Closed Flat Loop Conductor with Radius $2.0 \mathrm {~m}$ Is Located in a Changing Uniform Magnetic Field