Question

A circular loop has a radius of .3 meters a magnetic field has an initial strength of 2 tesla directed out of the page. over the course of 5 seconds the magnetic field is changed to 3 tesla directed out of the page calculate the induced voltage in the loop

Answer 1

Hi there!

We can use Faraday's Law to solve:

$\epsilon = N\frac{d\Phi _B}{dt}$

ε = Induced emf (? V)

N = Number of loops (1 loop)

ΦB = Magnetic flux (Wb)

We know that:

$\Phi_B = \oint B \cdot dA = B\cdot A$

Since the area of the loop remains the same, we can take this out of the time derivative.

We get:
$\frac{d\Phi_B}{dt} = A * \frac{dB}{dt}$

Also, since N = 1, we can now rewrite the equation for the induced emf as:
$\epsilon = A * \frac{dB}{dt}$

dB/dt is equivalent to the change in the magnetic field with respect to time:
$\Delta B = \frac{B_f - B_i}{\Delta t}\\\\\Delta B = \frac{3 - 2}{5} = 0.2 \frac{T}{s}$

Now, substitute this value into the equation for induced emf:
$\epsilon = \pi (0.3^2) * (0.2) = \boxed{0.0565 V}$

**Also, since the magnetic field INCREASED out of the page, this change in magnetic flux will create an induced CLOCKWISE current that produces a magnetic field into the page in order to oppose the increase in magnetic flux OUT of the page.