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Chapter 29: Problem 20

A metal hoop is laid flat on the ground. A magnetic field that is directed upward, out of the ground, is increasing in magnitude. As you look down on the hoop from above, what is the direction of the induced current in the hoop?

Short Answer

Expert verified
Answer: The induced current in the hoop is clockwise.

Step by step solution

01

Understand Faraday's Law

Faraday's Law of electromagnetic induction states that the electromotive force (EMF) induced in a closed loop is equal to the negative rate of change of magnetic flux through the loop. Mathematically, it is expressed as: \(\text{EMF} = - \frac{d \Phi_B}{dt}\) Where \(\Phi_B\) is the magnetic flux and \(t\) is time. The induced current in the hoop will create a magnetic field that opposes the change in the external magnetic field, as stated by Lenz's law.
02

Determine the direction of the change in magnetic flux

In this case, the magnetic field is increasing in magnitude and is directed upward, out of the ground. This means that the change in magnetic flux is also upward.
03

Apply the right-hand rule

To determine the direction of the induced current, we can use the right-hand rule. The right-hand rule states that if we point our thumb in the direction of the magnetic field and curl our fingers, the direction of our fingers represents the direction of the induced current. Since the change in magnetic flux is upward, point the thumb of your right hand upward. Curl your fingers around the hoop, and the direction in which your fingers curl is the direction of the induced current.
04

Determine the direction of the induced current

From Step 3, when pointing your right thumb upward, your fingers will curl around the hoop in a clockwise direction (when looking down at the hoop from above). Therefore, the induced current in the hoop is clockwise.

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Most popular questions from this chapter

An elastic circular conducting loop expands at a constant rate over time such that its radius is given by \(r(t)=r_{0}+v t\), where \(r_{0}=0.100 \mathrm{~m}\) and \(v=0.0150 \mathrm{~m} / \mathrm{s} .\) The loop has a constant resistance of \(R=12.0 \Omega\) and is placed in a uniform magnetic field of magnitude \(B_{0}=0.750 \mathrm{~T}\), perpendicular to the plane of the loop, as shown in the figure. Calculate the direction and the magnitude of the induced current, \(i\), at \(t=5.00 \mathrm{~s}\).

A square conducting loop with sides of length \(L\) is rotating at a constant angular speed, \(\omega\), in a uniform magnetic field of magnitude \(B\). At time \(t=0\), the loop is oriented so that the direction normal to the loop is aligned with the magnetic field. Find an expression for the potential difference induced in the loop as a function of time.

A wedding ring (of diameter \(2.0 \mathrm{~cm}\) ) is tossed into the air and given a spin, resulting in an angular velocity of 17 rotations per second. The rotation axis is a diameter of the ring. Taking the magnitude of the Earth's magnetic field to be \(4.0 \cdot 10^{-5} \mathrm{~T}\), what is the maximum induced potential difference in the ring?

A 100 -turn solenoid of length \(8 \mathrm{~cm}\) and radius \(6 \mathrm{~mm}\) carries a current of 0.4 A from right to left. The current is then reversed so that it flows from left to right. By how much does the energy stored in the magnetic field inside the solenoid change?

A rectangular conducting loop with dimensions \(a\) and \(b\) and resistance \(R\), is placed in the \(x y\) -plane. A magnetic field of magnitude \(B\) passes through the loop. The magnetic field is in the positive \(z\) -direction and varies in time according to \(B=B_{0}\left(1+c_{1} t^{3}\right),\) where \(c_{1}\) is a constant with units of \(1 / \mathrm{s}^{3}\) What is the direction of the current induced in the loop, and what is its value at \(t=1 \mathrm{~s}\) (in terms of \(a, b, R, B_{0},\) and \(\left.c_{1}\right) ?\)
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