Induced current and torque in a rectangular metal loop

In summary, we discussed a metallic rectangular loop mounted in an assembly that allows it to rotate, with a uniform magnetic field applied. We determined the magnetic flux threading the loop and found expressions for the induced current and power dissipated in the loop when rotated at a constant angular velocity. Finally, we considered the magnetic moment of the loop and found an expression for the torque needed to sustain the rotation at constant angular velocity.
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Homework Statement



(a) The metallic rectangular loop of length b and width a (ABCD) is mounted in an assembly, which allows it to be rotated about an axis bisecting AD and BC. The plane of the loop makes an angle θ to the horizontal and a uniform magnetic field B is applied vertically upwards (attached is a diagram). Write down an expression for the magnetic flux threading the loop.

(b) The loop is rotated at angular velocity ω such that θ = ωt. Given that the resistance of the wire around the loop is R, find an expression for the induced current in the loop at time t. Hence find the power dissipated in the loop.

(c) By considering the magnetic moment of the loop, find an expression for the torque that must be applied to the loop in order to sustain the rotation at constant angular velocity.


Homework Equations



No equations were given. However, useful equations for the questions might be:

(a) Magnetic flux = ∫B.dA

(b) IR = Change in magnetic flux / Change in time and P = I2R

(c) m = I A and T = IaB(bsinθ) = IABsinθ (where A is the area of the metal loop)

The Attempt at a Solution



(a) Magnetic Flux = ab B cosθ

(b) IR = d (ab cosωt B) / dt

I = (1/R) . d (ab cosωt B) / dt

Subbing this into: P = I2R,

P = (1/R) . (d (ab cosωt B) / dt)2

(c) I wasn't particularly sure where to start on this question, other than this magnetic moment being: m = I ab


Any help on this would be much appreciated, especially part (c), but please correct me on everything else if it's wrong :smile:
 

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  • #2
(a) is OK.
(b) you should perform the differentiation and get a closed form expressions for current and power.
(c) (torque) = (mag. moment) x B
work = integrated torque over angle (suggest 360 deg.)
work = avg. power computed in (b) x time. Suggest time of 1 rotation of 360 deg.
equate the two

vectors in bold
 

1. What is induced current in a rectangular metal loop?

Induced current is the flow of electric charge that occurs in a rectangular metal loop when it is placed in a changing magnetic field. This current is caused by Faraday's Law of Induction, which states that a changing magnetic field can induce an electric current in a conductor.

2. How is induced current related to torque in a rectangular metal loop?

The induced current in a rectangular metal loop creates a magnetic field that interacts with the external magnetic field, resulting in a torque on the loop. This is known as the torque equation and can be calculated by multiplying the current, the length of the loop, and the external magnetic field.

3. How does the shape of a rectangular metal loop affect the induced current and torque?

The shape of a rectangular metal loop can affect the induced current and torque in several ways. A larger loop will have a stronger induced current and a larger torque, while a smaller loop will have a weaker induced current and a smaller torque. Additionally, the orientation of the loop in relation to the external magnetic field can also impact the induced current and torque.

4. What factors can affect the magnitude of induced current and torque in a rectangular metal loop?

The magnitude of induced current and torque in a rectangular metal loop can be affected by several factors, including the strength and direction of the external magnetic field, the size and shape of the loop, and the material of the loop. The rate of change of the magnetic field and the resistance of the loop can also impact the magnitude of these values.

5. How is the direction of induced current and torque determined in a rectangular metal loop?

The direction of induced current and torque in a rectangular metal loop is determined by the right-hand rule. This rule states that if you point your right thumb in the direction of the external magnetic field, and your fingers in the direction of the current in the loop, your palm will face the direction of the induced torque. The direction of the induced current can be determined using the same rule, with the thumb pointing in the direction of the changing magnetic field.

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