# What is the magnitude of the emf induced in the loop and the direction of the current through R?

## The magnetic flux through the loop shown in the Fig. increases according to the relation phiB = 6t^2 + 7t, where capital phiB is in milliwebers and t is in seconds. What is the magnitude of the emf induced in the loop when t=2s? What is the direction of the current through R?

Apr 6, 2017

#### Answer:

$\left\mid m a t h c a l E \left(t = 2\right) \right\mid = 31 \text{ mV}$

Conventional (positive charged) current through the resistor is right to left.

#### Explanation:

First, in terms of the units, from Wiki (expurgated, and my math stuff added): "A change in magnetic flux ($\Delta {\Phi}_{B}$) of one weber ($W$) per second will induce an electromotive force ($m a t h c a l E$) of one volt".

Magnitude of EMF

Faraday's Law for magnetically induced currents:

$m a t h c a l E = - \frac{d {\Phi}_{B}}{\mathrm{dt}}$

${\Phi}_{B} \left(t\right) = 6 {t}^{2} + 7 t \text{ mW" = 10^(-3) (6t^2 + 7t ) " W}$

$m a t h c a l E \left(t = 2\right) = - {10}^{- 3} {\left(12 t + 7\right)}_{t = 2} = - 31 \cdot {10}^{- 3} \text{ V}$

$\left\mid m a t h c a l E \left(t = 2\right) \right\mid = 31 \text{ mV}$

Direction of Current

The green dots are arrowheads and indicate that the B field is moving "out of the page". And ${\Phi}_{B} \left(t\right) = 6 {t}^{2} + 7 t$ means that that is the direction in which the B field is increasing.

By Lenz's Law, the induced current will produce a back field opposing that field.

There are numerous ways to turn that into a current. I prefer the right hand thumb rule - really keeps things simple.

If we grasp the coil with thumb pointing in direction of pink arrow, our curled fingers suggest that the induced field inside the ring is also pointing "out of the page". That cannot be so. This extra magnetic field would just create more current, and more energy, and we have a super-perpetual motion machine.

If we grasp the coil with thumb pointing in direction of the next pink arrow, our curled fingers suggest that the induced field inside the ring is pointing " into the page", opposing the field that is creating the EMF. Must be right :)

So current through the resistor (conventional current, that is) is right to left.