A 100 A current circulates around a 2.0-mm-diameter superconducting ring. What is the on-axis magnetic field strength 5.0 cm from the ring?

검증된 단계별 안내

Determine the radius of the superconducting ring. Since the diameter is given as 2.0 mm, the radius \( r \) is half of the diameter: \( r = \frac{2.0 \text{ mm}}{2} = 1.0 \text{ mm} = 1.0 \times 10^{-3} \text{ m} \).

Use the Biot-Savart law to calculate the magnetic field on the axis of a current loop. The formula for the magnetic field at a point along the axis of a circular current loop is: \( B = \frac{\mu_0 I r^2}{2 (r^2 + z^2)^{3/2}} \), where \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \ \text{T·m/A} \)), \( I \) is the current, \( r \) is the radius of the loop, and \( z \) is the distance from the center of the loop along the axis.

Substitute the known values into the formula: \( I = 100 \ \text{A} \), \( r = 1.0 \times 10^{-3} \ \text{m} \), and \( z = 5.0 \ \text{cm} = 0.05 \ \text{m} \).

Simplify the expression for \( B \) by calculating the terms \( r^2 \), \( z^2 \), and \( (r^2 + z^2)^{3/2} \). Then multiply by the constants \( \mu_0 \), \( I \), and \( r^2 \) as per the formula.

After simplifying, the result will give the magnetic field strength \( B \) at a distance of 5.0 cm from the ring along its axis.

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주요 개념

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Superconductivity

Superconductivity is a phenomenon where certain materials exhibit zero electrical resistance and expel magnetic fields when cooled below a critical temperature. In superconductors, electric current can flow indefinitely without energy loss, making them ideal for applications like magnetic levitation and powerful electromagnets.

Magnetic Field of a Current Loop

The magnetic field generated by a current-carrying loop can be calculated using the Biot-Savart law or Ampère's law. For a circular loop, the magnetic field strength at a point along the axis can be determined by considering the contributions from each segment of the loop, which combine to create a net magnetic field.

Biot-Savart Law

The Biot-Savart law describes how electric currents produce magnetic fields. It states that the magnetic field dB at a point in space due to a small segment of current-carrying wire is directly proportional to the current and inversely proportional to the square of the distance from the wire segment to the point, factoring in the angle between the current direction and the line connecting the wire to the point.