# Average power dissipation for induced current

• auctor
In summary, a wire in the shape of a semi-circle rotates about an axis with constant angular velocity in a uniform magnetic field. The total resistance of the circuit is R. Neglecting the effects of the magnetic field generated by the current, the mean thermal power dissipated during one rotation period can be found using the equations for emf, flux, angle, current, and power. The average power is given by P=[ωBπa^2]^2/(8R), which is different from the answer given in the book. However, further analysis shows that the book's answer corresponds to a variation of the projection area of the loop, which may be a typo or a subtlety not mentioned in the given information.
auctor

## Homework Statement

A wire shaped as a semi-circle of radius a rotates about the axis OO’ with a constant angular velocity ω in a uniform magnetic field with induction B (attached figure). The plane of the rectangular loop is perpendicular to the magnetic field direction. The total resistance of the circuit is R. Neglecting effects of the magnetic field that is generated by the current in the circuit, find the mean thermal power dissipated during one rotation period.

## Homework Equations

Emf: ε=-dΦ/dt
Flux: dΦ=B⋅dS
Angle: ϑ=ωt
Current: I=ε/R
Power: P=εI
0 sin2(x)dx = 1/2

## The Attempt at a Solution

The area of the projection of the loop onto the direction perpendicular to the magnetic field changes according to S=Srectangle+πa2/2 cos(ϑ)=Srectangle+πa2/2 cos(ωt)
The magnetic induction isn't changing.
The flux as a function of time is then Φ=B(Srectangle+πa2/2 cos(ωt))
The emf is ε=-dΦ/dt = ωBπa2/2 sin(ωt)
The current is I=ωBπa2/(2R) sin(ωt)
The instantaneous power is P=[ωBπa2]2/(4R) sin2(ωt)
The average power is P=[ωBπa2]2/(8R)

However, the answer the book gives is P=[ωBπa2]2/(2R). This result would correspond to a variation of the projection area S=Srectangle+πa2 cos(ϑ). Not sure why this would be the case...

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Your work looks correct to me. I agree with your result for the average power.

(Your expression for the instantaneous power is missing a square on the sine function.)

Thanks! (Square for sine corrected.) Just wondered if the book has a typo or if I was missing some subtlety.

## What is average power dissipation for induced current?

The average power dissipation for induced current is the amount of power that is lost or converted to heat when an electric current is induced in a conductor. This power is dissipated due to the resistance of the conductor and is typically measured in watts.

## How is average power dissipation for induced current calculated?

The average power dissipation for induced current is calculated by multiplying the square of the current by the resistance of the conductor. This is also known as Joule's Law, which states that power dissipation is directly proportional to the square of the current and the resistance.

## What factors affect the average power dissipation for induced current?

Several factors can affect the average power dissipation for induced current, including the magnitude of the induced current, the resistance of the conductor, and the frequency of the induced current. Additionally, the length and thickness of the conductor can also impact the power dissipation.

## Why is average power dissipation for induced current important?

Average power dissipation for induced current is an important concept in understanding the efficiency of electrical systems. It helps engineers and scientists determine the amount of energy that is lost as heat, which can affect the overall performance and design of electrical devices.

## How can average power dissipation for induced current be reduced?

To reduce average power dissipation for induced current, one can use conductors with lower resistance or increase the frequency of the induced current. Additionally, using materials with higher conductivity can also help reduce power dissipation. Proper insulation and cooling techniques can also play a role in reducing power dissipation.

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