Problem 21 - College Physics

Question

A very thin 15.0 \(\mathrm{cm}\) copper bar is aligned horizontally along the east-west direction. If it moves horizontally from south to north at 11.5 \(\mathrm{m} / \mathrm{s}\) in a vertically upward magnetic field of \(1.22 \mathrm{T},\) (a) what potential difference is induced across its ends, and (b) which end (east or west) is at a higher potential? (c) What would be the potential difference if the bar moved from east to west instead?

Step-by-Step Solution

Answer

(a) 2.10 V; (b) East end is higher; (c) 0 V.

Step 1: Understand Faraday's Law

To determine the potential difference induced across the copper bar, we start by understanding that the motion of the bar in a magnetic field induces an electromotive force (EMF) based on Faraday's Law of Induction. The formula for the induced EMF is given by \( \varepsilon = B \cdot v \cdot L \), where \( B \) is the magnetic field strength, \( v \) is the velocity of the conductor, and \( L \) is the length of the conductor.

Step 2: Calculate Induced Voltage for South-North Movement

For part (a), the values are \( B = 1.22 \, \mathrm{T} \), \( v = 11.5 \, \mathrm{m/s} \), and \( L = 0.15 \, \mathrm{m} \). Substitute these into the EMF formula: \( \varepsilon = 1.22 \times 11.5 \times 0.15 \). Calculate the result to find the potential difference induced across the ends.

Step 3: Evaluate Direction and Sign of Induced Voltage

To determine which end is at a higher potential, use the right-hand rule. Point your thumb in the direction of the velocity (north), fingers in the direction of the magnetic field (up), and your palm will face the direction of the induced EMF. This shows that the magnetic force pushes positive charges toward the east end, making it higher in potential.

Step 4: Determine Effect of East-West Movement on Voltage

In part (c), the bar moves east to west. In this case, since the velocity is perpendicular to the previous direction, use the formula \( \varepsilon = B \cdot v \cdot L \) again. However, the setup is now different; calculate \( \varepsilon = 0 \) because the velocity is parallel to the magnetic field lines. So no additional potential difference is induced when it moves east-west as opposed to north.

Key Concepts

Electromotive Force (EMF)

Electromotive force (EMF) often refers to the voltage generated by a battery or a changing magnetic field. When a conductor, like a copper bar, moves through a magnetic field, it cuts through the magnetic lines of force. This action induces an EMF within the conductor.
This EMF can be calculated using the formula:

• \( \varepsilon = B \cdot v \cdot L \)

where:

• \( B \) is the magnetic field strength in Teslas,
• \( v \) is the velocity of the conductor in meters per second,
• \( L \) is the length of the conductor in meters.

The induced EMF is essentially the potential difference between the ends of the bar. This concept is a key principle in understanding electric generators and many other electromagnetic devices.

Magnetic Field

A magnetic field is a region around a magnetic material or a moving electric charge within which the force of magnetism acts. In this exercise, the field is described as vertically upward with a strength of 1.22 Tesla (T).
This magnetic field interacts with the moving copper bar, inducing an EMF due to the magnetic flux change through the bar.
Magnetic fields are vector fields, which means they have both magnitude and direction. They are often visualized by magnetic field lines. When a conductor moves through these field lines, electromagnetic induction occurs, creating an electrical potential difference across the conductor.

Right-Hand Rule

The right-hand rule is a valuable mnemonic in electromagnetism to determine the direction of the induced EMF. To apply it:

• Thumb: Point in the direction of the velocity of the moving conductor.
• Fingers: Point in the direction of the magnetic field.
• Palm: Faces the direction of the force exerted on positive charges.

In the scenario where the bar moves south to north, and the magnetic field is upward, the right-hand rule indicates that the electric force is toward the east. Consequently, the east end is at a higher potential, enabling us to determine the side with greater voltage without extensive calculations.

Induced Voltage

Induced voltage relates to the voltage generation across the ends of a conductor when it interacts with a magnetic field. This process is captured by Faraday's Law of Induction.
For the copper bar in the exercise, moving the bar south to north in an upward magnetic field resulted in an induced voltage. Here, the velocity, magnetic field strength, and length combine to produce a measurable potential difference. The formula:

• \( \varepsilon = 1.22 \times 11.5 \times 0.15 \)

calculates this induced voltage based on the given parameters. This simple yet powerful physics principle powers many technologies, from simple generators to complex power grids.

Lorentz Force

The Lorentz force is the combination of magnetic and electric forces on a point charge due to electromagnetic fields. When a charged particle such as an electron moves through a magnetic field, it experiences this force.
The force is given by:

• \( \mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \)

where:

• \( q \) is the charge,
• \( \mathbf{E} \) is the electric field,
• \( \mathbf{v} \) is the velocity,
• \( \mathbf{B} \) is the magnetic field.

For the copper bar moving across the field, the Lorentz force is responsible for pushing charges to one end of the bar, causing charge separation and thereby creating an EMF. This separation of charges results in the observable potential difference that was calculated in the exercise.