Calculate the potential difference induced in the circuit as a function of time

[blueyellow]

Homework Statement

A square circuit with side L lies in the x-y plane, is centred on the origin and has sides parallel to the x and y axes. At time t=0 it starts turning about the x axis, in a region characterised by a uniform magnetic field B parallel to z, with uniform angular speed omega. By considering the angle between the surface normal and the field, calculate the potential difference (electromotive force) induced in the circuit as a function of time.

Homework Equations

phi(subscript B)=BA cos theta

potential difference=-d(phi (subscript b))/dt

The Attempt at a Solution

potential difference=-d(BA cos theta)/dt

this isn't leading anywhere. I don't know how to differentiate it with respect to t

 

[Pi-Bond]

You need to express the angle theta (angle between surface normal and field lines) in terms of time. (theta starts from 0 degrees)

 

[blueyellow]

thank you, Pi-bond

potential difference=-d(BA cos omega*t)/dt

=BA omega sin (omega*t)

correct?

 

[Pi-Bond]

Yes, that is correct.

 

[rwisz]

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As a scientist, it is important to approach problems with a systematic and logical approach. In this case, we can start by breaking down the problem into smaller parts and identifying any relevant equations or principles that can help us solve it.

Firstly, we can recognize that the potential difference induced in the circuit is a result of Faraday's law of induction, which states that a changing magnetic field can induce an electric field. In this case, the changing magnetic field is due to the rotation of the circuit.

Next, we can use the equation for magnetic flux, phi(subscript B)=BA cos theta, where B is the magnetic field, A is the area of the circuit, and theta is the angle between the surface normal and the magnetic field.

Since the circuit is rotating about the x axis, the angle between the surface normal and the magnetic field will be changing with time. We can express this angle as a function of time, theta(t)=omega*t, where omega is the angular speed.

Substituting this into the equation for magnetic flux, we get phi(subscript B)=BA cos (omega*t).

Finally, we can use the equation for potential difference induced by a changing magnetic flux, potential difference=-d(phi(subscript B))/dt.

Taking the derivative with respect to time, we get potential difference=-BA omega sin (omega*t).

Therefore, the potential difference induced in the circuit as a function of time is given by the equation potential difference=-BA omega sin (omega*t). This shows that the potential difference will vary sinusoidally with time, with a maximum value of BA omega and a minimum value of -BA omega.

In conclusion, by considering the angle between the surface normal and the magnetic field, we were able to calculate the potential difference induced in the circuit as a function of time using principles from electromagnetism. This approach can be applied to other problems in physics, where breaking down the problem and using relevant equations and principles can help us find a solution.