A circular conducting ring with radius r0=0.0420 m lies in the x y- plane in a region of unifor

step 1 So we need to apply Faraday's law to calculate the magnitude and direction of the induced EMF. S

A circular conducting ring with radius r0=0.0420 m lies in...

A circular conducting ring with radius $r_{0}=0.0420 \mathrm{~m}$ lies in the $x y-$ plane in a region of uniform magnetic field $\overrightarrow{\boldsymbol{B}}=B_{0}\left[1-3\left(t / t_{0}\right)^{2}+2\left(t / t_{0}\right)^{3}\right] \hat{\boldsymbol{k}} .$ In this expression, $t_{0}=0.0100 \mathrm{~s}$ and is constant, $t$ is time, $\hat{k}$ is the unit vector in the $+z$ -direction, and $B_{0}=0.0800 \mathrm{~T}$ and is constant. At points $a$ and $b$ (Fig. $\mathbf{P 2 9 . 5 6}$ ) there is a small gap in the ring with wires leading to an external circuit of resistance $R=12.0 \Omega$. There is $\mathrm{no}$ magnetic field at the location of the external circuit. (a) Derive an expression, as a function of time, for the total magnetic flux $\Phi_{B}$ through the ring. (b) Determine the emf induced in the ring at time $t=5.00 \times 10^{-3} \mathrm{~s}$. What is the polarity of the emf? (c) Because of the internal resistance of the ring, the current through $R$ at the time given in part $(b)$ is only $3.00 \mathrm{~mA}$. Determine the internal resistance of the ring. (d) Determine the emf in the ring at a time $t=1.21 \times 10^{-2} \mathrm{~s}$. What is the polarity of the emf? (e) Determine the time at which the current through $R$ reverses its direction.

Key Concepts

Time-dependent Magnetic Fields

Time-dependent magnetic fields, which vary according to explicit functions of time, are central to many problems in electromagnetic induction. The rate at which a magnetic field changes directly affects the magnitude and polarity of the induced emf, making the temporal aspect of magnetic fields critical for predicting the behavior of induced currents.

Lenz's Law

Lenz's Law determines the direction of the induced current resulting from a change in magnetic flux. It posits that the induced emf generates a current whose magnetic field opposes the change in the original magnetic flux, ensuring energy conservation in the process of induction.

Induced Emf and Circuit Response

The induced electromotive force (emf) in a circuit due to a changing magnetic environment gives rise to a current, which is influenced by both the induced emf and the total resistance in the circuit. Analyzing the relationship between induced emf, current, and resistance is crucial for understanding how real-world circuits behave under time-varying magnetic fields.

Magnetic Flux

Magnetic flux is the measure of the magnetic field passing perpendicularly through a given area. It is defined as the integral of the magnetic field over the surface and quantifies how much of the field 'flows' through that area. Studying magnetic flux is essential for understanding induction phenomena in electromagnetic systems.

Faraday's Law of Induction

Faraday's Law connects a changing magnetic flux to the induced electromotive force (emf) in a circuit. Specifically, the law states that the induced emf in a closed loop is equal to the negative rate of change of the magnetic flux through the loop. This principle underlies the operation of transformers, generators, and various electromagnetic devices.

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