Problem 11 - University Physics with Modern Physics

Question

In a region of space, a magnetic field points in the +\(x\)-direction (toward the right). Its magnitude varies with position according to the formula \(B_x = B_0 + bx\), where \(B_0\) and \(b\) are positive constants, for \(x \geq\) 0. A flat coil of area \(A\) moves with uniform speed \(v\) from right to left with the plane of its area always perpendicular to this field. (a) What is the emf induced in this coil while it is to the right of the origin? (b) As viewed from the origin, what is the direction (clockwise or counterclockwise) of the current induced in the coil? (c) If instead the coil moved from left to right, what would be the answers to parts (a) and (b)?

Step-by-Step Solution

Answer

(a) \( Abv \), clockwise for right-left. If left-right, (a) \( Abv \), counterclockwise.

Step 1: Identify necessary formula

The formula for electromotive force (emf) induced in a coil moving in a magnetic field is given by Faraday's law of electromagnetic induction: \( \mathcal{E} = -\frac{d\Phi_B}{dt} \), where \( \Phi_B \) is the magnetic flux.

Step 2: Calculate magnetic flux through the coil

The magnetic flux \( \Phi_B \) through the coil is calculated as \( \Phi_B = B_x \cdot A \). Here, \( B_x = B_0 + bx \) is the magnetic field and \( A \) is the coil's area. Thus, \( \Phi_B = (B_0 + bx)A \).

Step 3: Determine the rate of change of flux

Since flux \( \Phi_B \) is \( (B_0 + bx)A \), the change in flux with respect to time as the coil moves is: \[ \frac{d\Phi_B}{dt} = A\frac{d(B_0 + bx)}{dt} = Ab\frac{dx}{dt} = Abv \], where \( \frac{dx}{dt} \) is the speed \( v \) of the coil.

Step 4: Calculate the induced emf

Using Faraday's law, the induced emf is \( \mathcal{E} = - Abv \). The negative sign indicates the direction according to Lenz's law, but the magnitude of the emf is \( Abv \).

Step 5: Determine the current direction for right-moving coil

When viewed from the origin and the coil is moving from right to left, the induced current, due to Lenz's law, will be in the clockwise direction to oppose the decrease in flux as it moves away from the origin.

Step 6: Reevaluating for coil moving left-to-right

If the coil moves from left to right, the expressions derived for emf magnitude remain the same: \( \mathcal{E} = Abv \).

Step 7: Determine the current direction for left-moving coil

Moving from left to right towards the origin, the induced current will be counterclockwise as viewed from the origin. This is because as the coil enters regions of stronger magnetic field, it opposes the increase in magnetic flux.

Key Concepts

Faraday's Law

Faraday's Law is a fundamental principle of electromagnetism which states that a change in magnetic flux through a coil will induce an electromotive force (emf) in the wire. This phenomenon is called electromagnetic induction. The law is mathematically represented by the formula:

• \( \mathcal{E} = -\frac{d\Phi_B}{dt} \)

where \( \mathcal{E} \) is the induced emf and \( \Phi_B \) is the magnetic flux. Faraday's Law highlights the relationship between the rate of change of magnetic flux and the resulting emf.
This change can be due to alterations in the magnetic field strength, the area of the coil, or its orientation relative to the magnetic field. Negative sign in Faraday's Law comes from Lenz's law, explaining the direction of the induced current.
In practical terms, Faraday's Law forms the basis for electric generators, transformers, and many other technologies that involve changing magnetic fields to generate electricity.

Magnetic Flux

Magnetic flux, denoted as \( \Phi_B \), is a measure of the number of magnetic field lines that pass through a given area. It's an important concept when dealing with electromagnetic induction as it determines the magnitude of induced emf according to Faraday's Law. The magnetic flux through a surface is calculated as:

• \( \Phi_B = B \cdot A \cdot \cos(\theta) \)

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 perpendicular to the surface.
When a coil moves in a non-uniform magnetic field, changes in magnetic flux can be the result of:

• changes in \( B \)
• the coil's position \( x \)
• the orientation of the coil relative to the field.

In the given exercise, the magnetic flux was calculated using the expression \( \Phi_B = (B_0 + bx)A \) showing how flux varies with the coil's position in a non-uniform field. This impacts the induced current based on how the coil moves relative to the changing field strength.

Lenz's Law

Lenz's Law describes the directionality of induced currents. It's essentially a manifestation of the conservation of energy principle, ensuring that the induced current will always oppose the motion or change causing it. In mathematical terms, this is represented as the negative sign in Faraday's Law:

• \( \mathcal{E} = -\frac{d\Phi_B}{dt} \)

Lenz's Law states that the direction of any induced current will be such that the magnetic field it creates will oppose the change in the original magnetic flux. This means:

• If a magnetic field through a coil increases, the induced current produces its own magnetic field opposing this increase.
• Conversely, if the magnetic field through a coil decreases, the induced current generates a magnetic field trying to maintain it.

In the textbook example, when the coil moves from right to left (as viewed from the origin), the magnetic flux through the coil decreases as the coil moves away, and Lenz’s Law predicts a clockwise induced current, trying to counteract this decrease. Conversely, moving from left to right causes an increase in flux, leading to a counterclockwise current.