A 13.0-cm-diameter circular loop of wire is placed with the plane of the loop parallel to the uniform magnetic field between the pole pieces of a large magnet. When 3.80 A flows in the coil, the torque on it is 0.185 m N What is the magnetic field strength?

Verified step by step guidance

1. Convert the diameter of the circular loop into the radius. The radius is half the diameter: \( r = \frac{d}{2} \). Here, \( d = 13.0 \; \text{cm} \), so convert it to meters: \( r = \frac{13.0}{2} \times 10^{-2} \; \text{m} \).
2. Calculate the area of the circular loop using the formula for the area of a circle: \( A = \pi r^2 \). Substitute the radius \( r \) into this formula.
3. Recall the formula for the torque on a current-carrying loop in a magnetic field: \( \tau = n I A B \sin \theta \), where \( \tau \) is the torque, \( n \) is the number of turns in the loop (assume \( n = 1 \) unless stated otherwise), \( I \) is the current, \( A \) is the area, \( B \) is the magnetic field strength, and \( \theta \) is the angle between the magnetic field and the normal to the plane of the loop. Here, \( \sin \theta = 1 \) because the plane of the loop is parallel to the magnetic field.
4. Rearrange the formula to solve for the magnetic field strength \( B \): \( B = \frac{\tau}{n I A \sin \theta} \). Substitute \( \tau = 0.185 \; \text{m} \cdot \text{N} \), \( n = 1 \), \( I = 3.80 \; \text{A} \), \( A \) (calculated in step 2), and \( \sin \theta = 1 \) into the equation.
5. Simplify the expression to find the value of \( B \), the magnetic field strength. Ensure all units are consistent (e.g., meters, amperes, newtons) before performing the calculation.

Key Concepts

Torque in a Magnetic Field

Torque in a magnetic field is the rotational force experienced by a current-carrying loop when placed in a magnetic field. It is given by the formula τ = n * I * A * B * sin(θ), where τ is torque, n is the number of turns, I is the current, A is the area of the loop, B is the magnetic field strength, and θ is the angle between the magnetic field and the normal to the loop's plane.

Area of a Circular Loop

The area of a circular loop is calculated using the formula A = π * r², where r is the radius of the loop. For a loop with a diameter of 13.0 cm, the radius is 6.5 cm (0.065 m). This area is crucial for determining the torque when the loop is subjected to a magnetic field.

Magnetic Field Strength

Magnetic field strength, denoted as B, is a measure of the magnetic force experienced by a unit current in a magnetic field. It is expressed in teslas (T) and can be calculated using the rearranged torque formula, allowing us to find B when the torque, current, and area of the loop are known.