Problem 13 - College Physics

Question

A circular loop of wire is in a spatially uniform magnetic field, as shown in Figure \(21.51 .\) The magnetic field is directed into the plane of the figure. Determine the direction (clockwise or counterclockwise) of the induced current in the loop when (a) \(B\) is increasing; (b) \(B\) is decreasing; (c) \(B\) is constant with a value of \(B_{0}\) . Explain your reasoning.

Step-by-Step Solution

Answer

(a) Counterclockwise; (b) Clockwise; (c) No current.

Step 1: Understanding Faraday's Law of Induction

Faraday's Law states that a change in magnetic flux through a loop induces an electromotive force (EMF) in the loop. The induced current will flow in a direction such that the magnetic field it creates opposes the change in magnetic flux (Lenz's Law).

Step 2: Evaluating Case (a) - Increasing Magnetic Field

When the magnetic field \( B \) is increasing, the magnetic flux through the loop is also increasing. According to Lenz's Law, the induced current will circulate in such a direction as to create a magnetic field opposing the increase in flux. Since the external magnetic field is directed into the plane, to oppose this increase, the induced current must be counterclockwise (to produce a magnetic field out of the plane).

Step 3: Evaluating Case (b) - Decreasing Magnetic Field

When the magnetic field \( B \) is decreasing, the magnetic flux through the loop is decreasing. The induced current will circulate in a direction that tries to maintain the original flux (opposing the decrease). Since the external field is into the plane, the induced current must be clockwise to produce a magnetic field into the plane.

Step 4: Evaluating Case (c) - Constant Magnetic Field

If the magnetic field \( B \) is constant (\( B_0 \)), there is no change in magnetic flux through the loop. Therefore, according to Faraday's Law, no EMF is induced, and no current will flow in the loop.

Key Concepts

Lenz's Law

Lenz's Law is a crucial principle when working with electromagnetic induction. It tells us how the direction of an induced current in a loop is determined. The law is named after the Russian physicist Heinrich Lenz, who formulated it in the 19th century. According to Lenz's Law, an induced current will always flow in a direction such that its magnetic field opposes the change that produced it. This principle is essential in conserving energy, ensuring that the energy required to change the magnetic flux is not created out of thin air.

For example, if a magnetic field through a loop is increasing, the induced current will create its own magnetic field that opposes the increase. Conversely, if the magnetic field is decreasing, the induced current will try to maintain the magnetic field by supporting it. This is why, in the Original Exercise, when the field is increasing, the induced current is counterclockwise, and when the field is decreasing, it is clockwise.

Magnetic Flux

Magnetic flux is a measure of the number of magnetic field lines passing through a given area, such as a loop of wire. It is an integral concept when discussing electromagnetic induction because changes in magnetic flux over time induce electromotive force (EMF) according to Faraday's Law of Induction.

Mathematically, magnetic flux (\( \Phi \) ) is given by the formula:\[\Phi = B \times A \times \cos(\theta)\]where:

• \( B \) is the magnetic field strength,
• \( A \) is the area through which the field lines pass, and
• \( \theta \) is the angle between the field lines and the normal (perpendicular) to the surface.

In the exercise mentioned, the loop is always perpendicular to the magnetic field, so \( \theta = 0 \) making \( \cos(\theta) = 1 \). Variations in magnetic flux are what cause the induced currents in the Original Exercise scenarios because they prompt a change in the magnetic field density through the loop.

Electromotive Force (EMF)

Electromotive Force, often abbreviated as EMF, is the driving force behind the movement of electrons in a conductor. Despite its name suggesting a force, it is actually a potential difference created by changing magnetic flux, acting like a battery in the circuit. According to Faraday's Law, EMF (\( \mathcal{E} \) ) can be calculated as:\[\mathcal{E} = -\frac{d\Phi}{dt}\]The negative sign in the equation is indicative of Lenz's Law, demonstrating that the induced EMF always opposes the change in magnetic flux.

In the Original Exercise case scenarios:

• When the magnetic field strength \( B \) is increasing or decreasing, it means \( \frac{d\Phi}{dt} eq 0 \), thus producing a non-zero EMF, leading to induced currents.
• When \( B \) is constant, \( \frac{d\Phi}{dt} = 0 \), resulting in no induced EMF and thus no current.

Understanding EMF is essential because it translates changes in the magnetic environment of a circuit into measurable electrical effects.