Problem 11 - University Physics with Modern Physics

Question

In a region of space, a magnetic ficld points in the \(+x\) -direction (toward the right). Its magnitude varies with position according to the formula \(B_{x}=B_{0}+b x,\) 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) \( \varepsilon = -bAv \); (b) Clockwise current; (c) \( \varepsilon = -bAv \), Counterclockwise current.

Step 1: Understand the problem

We need to determine the induced electromotive force (emf) in a coil moving through a magnetic field that varies with position. The coil moves in such a way that its plane is always perpendicular to the magnetic field, and the field's strength depends on the x-position.

Step 2: Determine the formula for emf

The induced emf in a coil can be calculated using Faraday's Law of Electromagnetic Induction which states that the emf \( \varepsilon \) is equal to the negative change in magnetic flux \( \Phi \) over time: \( \varepsilon = -\frac{d\Phi}{dt} \). The magnetic flux \( \Phi \) is given by \( \Phi = B_x A \) where \( B_x = B_0 + bx \).

Step 3: Calculate the rate of change of magnetic flux

Since the coil moves with a constant velocity \( v \), the position of the coil changes as \( x = vt \). The magnetic flux becomes a function of time: \( \Phi(t) = (B_0 + bvt)A \). The rate of change of the flux is \( \frac{d\Phi}{dt} = bA \frac{dx}{dt} = bAv \).

Step 4: Apply Faraday's Law

Using Faraday's Law, the emf induced in the coil is \( \varepsilon = -\frac{d\Phi}{dt} = -bAv \). The negative sign indicates the direction of the induced emf (according to Lenz's law), which will oppose the motion that created it.

Step 5: Determine the direction of the current

As the coil is moving from right to left, the increasing magnetic field induces a current in such a direction as to oppose the increase. Viewed from the origin, using the right-hand rule, the induced current is clockwise.

Step 6: Consider reverse direction (left to right)

If the coil moves from left to right, the change in the magnetic field is still \( bAv \), but the direction of motion with respect to the field gradient is reversed. Faraday's Law would give the same magnitude of emf: \( \varepsilon = -bAv \). However, the direction of the induced current would be counterclockwise, viewing from the origin.

Key Concepts

Magnetic Field

The magnetic field is a vector field that exerts a force on moving electric charges and magnetic dipoles. In this exercise, the magnetic field is described in the positive x-direction, towards the right. Its magnitude varies with position according to the equation \(B_{x} = B_{0} + bx\), where \(B_{0}\) represents a constant base field strength and \(b\) indicates the rate at which the field strength changes with position \(x\).
The key points to remember about magnetic fields in this context are:

• The field is uniform in direction but varies in magnitude depending on position.
• The field strength is greater towards increasing values of \(x\).
• The field directly influences the induction of emf in moving coils within it.

Understanding these aspects of magnetic fields is crucial, as they affect how electromagnetic induction occurs when a conductive loop moves through them.

Induced EMF

Induced electromotive force (emf) is a crucial concept in electromagnetism, often explained via Faraday's Law. This law describes how an emf is generated in a loop when exposed to a changing magnetic flux. For this exercise, the formula for induced emf is \(\varepsilon = -\frac{d\Phi}{dt}\). Here, \(\Phi\) denotes the magnetic flux through the coil area, calculated as \(\Phi = B_{x}A\), with \(B_{x}\) being the magnetic field strength and \(A\) being the area of the coil.

Key points on induced emf include:

• Induced emf arises when there is a change in magnetic flux either due to the movement of a coil or a change in the field itself.
• The negative sign in Faraday's Law indicates the direction of the induced current (Lenz's Law), opposing the change in magnetic flux.
• In the exercise, as the coil moves with velocity \(v\), the emf is calculated as \(\varepsilon = -bAv\), showing dependence on the coil's speed and flux gradient.

Appreciating how changes in movement and field strength contribute to emf helps in visualizing how devices work at a fundamental level.

Lenz's Law

Lenz's Law offers an intuitive understanding of the direction of induced currents in electromagnetism. According to Lenz's Law, the direction of induced current is such that it opposes the change in the magnetic flux that produced it. This principle ensures the conservation of energy within electromagnetic systems.
In practical terms:

• If a moving coil experiences an increasing magnetic field, the induced current will flow in a direction that creates a counteracting magnetic field.
• In the exercise, as the coil moves left through an increasing field strength, the new magnetic field generated by the induced current flows clockwise as viewed from the origin.
• If the coil's movement is reversed (left to right), the induced current flows counterclockwise, maintaining opposition to the directional change in flux.

Lenz's Law is an essential tool for predicting how systems will behave when they are subject to varying magnetic conditions, ensuring engineers can design circuits that respond predictably.