A circular loop of wire of radius r is in a uniform magnetic...

A circular loop of wire of radius $r$ is in a uniform magnetic field, with the plane of the loop perpendicular to the direction of the field (Fig. P31.61). The magnetic field varies with time according to $B(t)=a+b t$, where $a$ and $b$ are constants. (a) Calculate the magnetic flux through the loop at $t=0 .$ (b) Calculate the emf induced in the loop. (c) If the resistance of the loop is $R$, what is the induced current? (d) At what rate is electrical energy being delivered to the resistance of the loop?

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Key Concepts

Electrical Power Dissipation

Electrical power dissipation in a resistor is the rate at which electrical energy is converted to heat due to the resistance. It can be calculated using the relationship P = I^2R or equivalently, P = V^2/R, where I is the current and V is the voltage across the resistor.

Ohm's Law

Ohm's law defines the relationship between voltage, current, and resistance in an electrical circuit. It implies that the current flowing through a conductor is directly proportional to the applied voltage (or induced emf) and inversely proportional to the resistance.

Faraday's Law of Induction

Faraday's law relates the induced electromotive force (emf) in a closed loop to the rate of change of the magnetic flux through the loop. It states that the induced emf is equal to the negative time derivative of the magnetic flux, embodying the concept that a changing magnetic environment generates an electric field.

Magnetic Flux

Magnetic flux quantifies the amount of magnetic field passing through a given area. It is calculated as the product of the magnetic field strength and the area the field penetrates, with the flux maximized when the field is perpendicular to the surface.

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