A coil 4.00 cm in radius, containing 500 turns, is placed in...

A coil 4.00 cm in radius, containing 500 turns, is placed in a uniform magnetic field that varies with time according to $B =$ (0.0120 T/s)$t$ + (3.00 $\times$ 10$^{-5}$ T/s$^4)t^4$. The coil is connected to a 600-$\Omega$ resistor, and its plane is perpendicular to the magnetic field. You can ignore the resistance of the coil. (a) Find the magnitude of the induced emf in the coil as a function of time. (b) What is the current in the resistor at time $t =$ 5.00 s?

Key Concepts

Faraday's Law of Electromagnetic Induction

Faraday's Law explains that a changing magnetic flux through a loop or coil induces an electromotive force (emf) in the circuit. The induced emf is given by the negative rate of change of the magnetic flux multiplied by the number of turns in the coil. This principle is used to relate the time variation of the magnetic field to the generated voltage in the coil.

Magnetic Flux

Magnetic flux is defined as the product of the magnetic field, the area through which the field lines pass, and the cosine of the angle between the field and the normal to the surface. In the context of coils or loops, calculating the magnetic flux is essential because its rate of change directly determines the induced emf when the magnetic field is time-dependent.

Time-Varying Magnetic Fields

Time-varying magnetic fields are those whose magnitude (or direction) changes with time. Such variations cause changes in the magnetic flux through a coil or loop, leading to the induction of emf according to Faraday's law. Understanding how the magnetic field varies with time is crucial to predicting and calculating the dynamics of the induced electrical response.

Ohm's Law in Resistive Circuits

Ohm's Law establishes the relationship between voltage, current, and resistance in an electrical circuit (V = IR). When an emf is induced in a coil connected to a resistor, Ohm's law is applied to determine the current flowing through the resistor. This concept is key in connecting the induced voltage from Faraday's law to the observable electrical current in the circuit.

Transcript

00:01 Okay, start with faraday's law in part a, which says that induced emf is equal to number of turns of the coil times rate of change of magnetic flux dfi dt.

00:15 And so this is n times rate of change or change with respect to time of phi b, which is b, the magnetic field strength, times a the area, times.

00:30 Cosines 0 because the field is perpendicular to the earth magnetic field.

00:34 The coil is perpendicular to the earth's magnetic field.

00:38 But a here is a constant so it comes out here.

00:41 And so this turns out to be n a rate of change to respect to time of the magnetic field strength b.

00:52 And so b, so d b, d t is dd t is dd t of um, dd of um, um, of this expression we're given for b which is 0 .012 t plus point 012 tesla per second t plus point no plus 3 times 10 to the negative 5 tesla over seconds to the fourth which is a unit um t to the fourth.

01:37 Okay, we're going to differentiate that with respect to t.

01:41 And so this comes out to be, so here you just have the t term.

01:46 So this is just 0 .1 .2, tesla per second.

01:52 Okay.

01:53 Plus here you have two of the fourth, so you multiply three times 10 negative 5 by 4, which is 1 .2 times 10 of the negative 4.

02:02 It tells up for a second to the fourth and you lose a single power of t so times t to the or t cubed.

02:19 So now that you have that, you have dbd t.

02:22 You want to know what n a is.

02:24 So a, and we're given.

02:29 A we're not given, but we are given a radius.

02:31 So assuming a circular coil, a is pi, our...