(II) A wire is formed into the shape of two half circles connected by equal-length straight sections as shown in Fig. 28–48. A current I flows in the circuit clockwise as shown. Determine (a) the magnitude and direction of the magnetic field at the center, C, and (b) the magnetic dipole moment of the circuit.

Verified step by step guidance

1. Step 1: Understand the geometry of the wire. The wire consists of two half-circles of radius R connected by two straight sections of equal length. The current flows clockwise through the circuit. The center of the wire, C, is at the center of the two half-circles.
2. Step 2: Calculate the magnetic field contribution at point C due to the two half-circles. Use the Biot-Savart law for a circular current loop. The magnetic field at the center of a full circular loop is given by: $B=\frac{\mu ₀I}{2R}$. Since each half-circle contributes half of this field, the magnetic field due to one half-circle is: $B$_half = μ₀I4R. Add the contributions from both half-circles, keeping in mind their directions.
3. Step 3: Analyze the magnetic field contribution from the straight sections. The straight sections of the wire do not contribute to the magnetic field at point C because the current in these sections is parallel to the line connecting them to point C. The Biot-Savart law indicates that the magnetic field contribution from such sections at the center is zero.
4. Step 4: Determine the net magnetic field at point C. Combine the contributions from the two half-circles. Since the current flows clockwise, the magnetic field contributions from both half-circles at point C will add up in the same direction. Use the formula derived in Step 2 to calculate the total magnetic field: $B$_total = 2 × μ₀I4R = μ₀I2R. The direction of the magnetic field can be determined using the right-hand rule.
5. Step 5: Calculate the magnetic dipole moment of the circuit. The magnetic dipole moment is given by: $\mu =IA$, where A is the area enclosed by the circuit. The total area enclosed by the circuit is the sum of the areas of the two half-circles: $A=\; 2\; \times \frac{\pi R2}{2}=\pi R2$. Substitute this into the formula for the magnetic dipole moment to find: $\mu =I\pi R2$.

Key Concepts

Magnetic Field Due to Current

The magnetic field generated by an electric current is described by Ampère's Law, which states that the magnetic field around a closed loop is proportional to the current flowing through it. For a circular loop, the magnetic field at the center can be calculated using the formula B = (μ₀I)/(2R), where μ₀ is the permeability of free space, I is the current, and R is the radius of the loop.

Superposition of Magnetic Fields

When multiple current-carrying conductors are present, the total magnetic field at a point is the vector sum of the magnetic fields produced by each conductor. This principle of superposition allows us to analyze complex circuits by calculating the magnetic field contributions from each segment separately and then combining them to find the resultant field.

Magnetic Dipole Moment

The magnetic dipole moment (μ) is a vector quantity that represents the strength and orientation of a magnetic source. For a current loop, it is defined as μ = I * A, where I is the current and A is the area of the loop. The direction of the dipole moment is given by the right-hand rule, pointing perpendicular to the plane of the loop in the direction of the current.