For the situation described in Figure P31.32, the magnetic...

For the situation described in Figure P31.32, the magnetic field changes with time according to the expression $B=\left(2.00 t^{3}-4.00 t^{2}+0.800\right) \mathrm{T}$, and $r_{2}=2 R=$ $5.00 \mathrm{~cm}$ (a) Calculate the magnitude and direction of the force exerted on an electron located at point $P_{2}$ when $t=2.00 \mathrm{~s}$. (b) At what time is this force equal to zero?

For the situation described in Figure P31.32, the magnetic field changes with time according to the expression $B=\left(2.00 t^{3}-4.00 t^{2}+0.800\right) \mathrm{T}$, and $r_{2}=2 R=$ $5.00 \mathrm{~cm}$
(a) Calculate the magnitude and direction of the force exerted on an electron located at point $P_{2}$ when $t=2.00 \mathrm{~s}$. (b) At what time is this force equal to zero?

Key Concepts

Lenz's Law

Lenz's law provides the direction of the induced electromotive force and electric field, stating that the induced effect opposes the change in magnetic flux that produced it. This concept is important for understanding the direction of the force acting on charged particles in systems with time-varying magnetic fields.

Time Derivative of Magnetic Field

Calculating the time derivative of a magnetic field function is crucial in determining how rapidly the field is changing. This derivative helps compute the rate of change of the magnetic flux, which directly influences the magnitude of the induced EMF and the corresponding induced electric field through Faraday's law.

Electric Force on a Charge

The electric force exerted on a charged particle is determined by the product of the charge and the electric field acting on it. For an electron, this force is calculated by multiplying its charge by the magnitude of the induced electric field, taking into account the sign of the electron's charge.

Induced Electric Field

In the presence of a changing magnetic field, an electric field is generated even in the absence of a material conductor. This induced electric field rotates around the region of the changing magnetic flux and is related to the induced EMF by the line integral of the electric field along a closed path.

Magnetic Flux

Magnetic flux is the measure of the total magnetic field passing through a given area. It is computed as the product of the magnetic field and the area perpendicular to the field. In problems involving time-varying magnetic fields, the rate of change of the flux is a key factor in determining the induced EMF.

Faraday's Law of Electromagnetic Induction

This principle states that a changing magnetic flux through a closed loop induces an electromotive force (EMF) in that loop. It is essential when dealing with time-varying magnetic fields, as it explains how the variations in the magnetic field create an electric field and thus a voltage in nearby conductive paths.

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