How Does a Changing Magnetic Field Affect Nearby Electric Fields?

[DeadFishFactory]

Homework Statement

The magnetic field is uniform and out of the page inside a circle of radius R, and is essentially zero outside the circular region (see the figure). The magnitude of the magnetic field is changing with time; as a function of time the magnitude of the magnetic field is (B0 + bt^3). r1 = 6.4 cm, r2 = 63 cm, B0 = 1.5 T, b = 1.4 T/s3, t = 0.8 s, and R = 15 cm.


(a) What is the direction of the "curly" electric field at location P, a distance r1 to the left of the center (r1 < R)?

(b) What is the magnitude of the electric field at location P? (Hint: remember that "emf" is the integral of the non-Coulomb electric field around a closed path.)

(c) What is the direction of the "curly" electric field at location Q, a distance r2 to the right of the center (r2 > R)?

(d) What is magnitude of the "curly" electric field at location Q?

Homework Equations

E∫dA = emf
emf=dΦB/dT
ΦB = BxA

The Attempt at a Solution

I have no clue. It says to use the E∫dA = emf, but emf is not given. That (B0+bt^3) is just confusing me.

 

[kuruman]

Faraday's Law says emf = -dΦ/dt. Does that help?

 

[DeadFishFactory]

Not much because I don't know what to do with it. What would the flux be? Would it be
emf = -dΦ/dt

Φ = BxA

emf = (-d/dt)BxA?

Does B = B0 + bt^3?

 

[kuruman]

Not much because I don't know what to do with it.

Then you need to read your textbook and learn about Faraday's Law.

What would the flux be? Would it be
emf = -dΦ/dt

Φ = BxA

emf = (-d/dt)BxA?

I would not put a cross in between because it implies a cross product and that's not we have here. Actually we have a dot product. You also need to learn about magnetic flux and how it is defined. Flux is not just "Field times area". Only the normal component to the area contributes to the flux.

Does B = B0 + bt^3?

Yes.

 

[rdl]

As a scientist, it is important to approach any problem with a clear and logical thought process. Let's break down the given information and equations to come up with a solution.

Firstly, we are given a magnetic field that is changing with time, with a magnitude of (B0 + bt^3). This means that the magnetic field is increasing with time, as t^3 is a cubic term and b is a positive constant. We are also given the values of r1, r2, B0, b, t, and R.

(a) To determine the direction of the "curly" electric field at location P, we can use the right-hand rule. Since the magnetic field is increasing with time, the direction of the electric field will be counterclockwise. This is because the change in magnetic field induces a non-Coulomb electric field that is perpendicular to it, according to Faraday's law.

(b) To calculate the magnitude of the electric field at location P, we can use the equation E∫dA = emf, where emf is the electromotive force, or the induced voltage. From Faraday's law, we know that emf is equal to the change in magnetic flux over time. In this case, the change in magnetic flux is equal to the change in magnetic field (B0 + bt^3) multiplied by the area of the circle, πr1^2. Therefore, we can write:

E∫dA = (B0 + bt^3)πr1^2/t = (1.5 + 1.4(0.8)^3)π(0.064)^2/0.8 = 0.047 T

(c) To determine the direction of the "curly" electric field at location Q, we can again use the right-hand rule. Since the magnetic field is increasing with time, the direction of the electric field will be clockwise, opposite to the direction at location P.

(d) To calculate the magnitude of the electric field at location Q, we can use the same equation as in part (b), but with the new values for r2 and t:

E∫dA = (B0 + bt^3)πr2^2/t = (1.5 + 1.4(0.8)^3)π(0.63)^2/0.8 = 0.232 T

In