The power cable for an electric trolley (Fig. 27–60) carries a horizontal current of 330 A toward the east. The Earth’s magnetic field has a strength 5.0 x 10^-5 T and makes an angle of dip of 22° at this location. Calculate the magnitude and direction of the magnetic force on a 15-m length of this cable.

Verified step by step guidance

1. Identify the relevant formula for the magnetic force on a current-carrying conductor in a magnetic field: \( F = I L B \sin \theta \), where \( F \) is the magnetic force, \( I \) is the current, \( L \) is the length of the conductor, \( B \) is the magnetic field strength, and \( \theta \) is the angle between the current direction and the magnetic field.
2. Determine the angle \( \theta \) between the current direction (east) and the magnetic field. The Earth's magnetic field makes an angle of dip of 22° with the horizontal, so the angle between the current and the magnetic field is \( 90° - 22° = 68° \).
3. Substitute the given values into the formula: \( I = 330 \; \text{A} \), \( L = 15 \; \text{m} \), \( B = 5.0 \times 10^{-5} \; \text{T} \), and \( \sin \theta = \sin 68° \). The formula becomes \( F = 330 \cdot 15 \cdot (5.0 \times 10^{-5}) \cdot \sin 68° \).
4. Calculate the magnitude of the magnetic force by evaluating the product of the terms in the formula. Ensure that the sine of the angle is computed correctly.
5. Determine the direction of the magnetic force using the right-hand rule. Point your fingers in the direction of the current (east), curl them toward the direction of the magnetic field (downward at an angle of 22° to the horizontal), and your thumb will point in the direction of the force. The force will be perpendicular to both the current and the magnetic field.

Key Concepts

Magnetic Force on a Current-Carrying Conductor

When a current flows through a conductor in a magnetic field, it experiences a magnetic force. This force can be calculated using the formula F = I * L * B * sin(θ), where F is the force, I is the current, L is the length of the conductor, B is the magnetic field strength, and θ is the angle between the current direction and the magnetic field.

Earth's Magnetic Field

The Earth's magnetic field is a vector field that extends from the Earth's interior out into space, characterized by its strength and direction. At any location, it can be described by its magnitude (measured in teslas) and its angle of dip, which indicates how steeply the field lines enter the Earth. Understanding the local magnetic field is crucial for calculating the forces on current-carrying conductors.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic force on a current-carrying conductor. By pointing the thumb of the right hand in the direction of the current and the fingers in the direction of the magnetic field, the palm will face the direction of the force. This rule helps visualize the interaction between current, magnetic field, and force.