Question

A magnetic field is aligned perpendicular to the plane of a circular loop of wire with radius 5.0 cm and 20 turns. The magnetic field B changes with time t according to the following function: B 12 exp(-3t) +1.5t +6 T Determine the induced emf in the circular loop of wire as a (i) function of time, and the time at which the induced emf is 0.1 V. (ii) If the magnetic fieldis instead aligned at 30° with respect to the normal of the circular loop of wire, find the induced emf as a function of time.

Answer 1

`V = 0.157(-36 e^(-3t) + 1.5)`

`t = 1.24 s`

Part b)

`EMF = 0.136(-36e^(-3t) + 1.5)`

Step-by-step explanation:

magnetic field due to external source is given as

`B = 12 e^(-3t) + 1.5 t + 6`

area of the loop is given as

`A = \pi r^2`

`A = \pi(0.05)^2`

`A = 7.85 * 10^(-3) m^2`

Now we have

`\phi = NBAcos0`

`\phi = (20)(12e^(-3t) + 1.5t + 6)(7.85 * 10^(-3))`

`V = (d\phi)/(dt)`

`V = 0.157(d)/(dt)(12e^(-3t) + 1.5t + 6)`

`V = 0.157(-36 e^(-3t) + 1.5)`

now we need to find the time at which voltage is 0.1 Volts so we have

`0.1 V = 0.157(-36e^(-3t) + 1.5)`

`t = 1.24 s`

Part b)

If magnetic field is inclined at an angle of 30 degree with the normal of the loop then

`\phi = NBAcos30`

now we know that induced EMF is given as

`EMF = (d\phi)/(dt)`

`EMF = NAcos30(dB)/(dt)`

`EMF = (20)(7.85 * 10^(-3))cos30((d)/(dt)(12e^(-3t) + 1.5t + 6))`

`EMF = 0.136(-36e^(-3t) + 1.5)`