Question

The wire in the figure carries a current \(i\) and contains a circular arc of radius \(R\) and angle \(\pi / 2\) and two straight sections that are mutually perpendicular and, if extended, would intersect the center, \(C,\) of the arc. What is the magnetic field at point \(C\) due to the wire? a) \(B=\frac{\mu_{0} i}{2 R}\) b) \(B=\frac{\mu_{0} i}{4 R}\) c) \(B=\frac{\mu_{0} i}{6 R}\) d) \(B=\frac{\mu_{0} i}{8 R}\) e) \(B=\frac{\mu_{0} i}{12 R}\)

Short answer

Answer: The magnetic field at the center of the circular arc, point C, is given by the expression \(B=\frac{\mu_{0} i}{8 R}\).

Step-by-step solution

Determine the magnetic field due to the circular arc

To find the magnetic field due to the circular arc, we can use the Biot-Savart Law in the form: \(d B = \frac{\mu_{0} I}{4 \pi r} d L \times \frac{\widehat{R}}{R^3}\). Since we know the circular arc has an angle of \(\pi / 2\) and a radius R, we can replace \(d L\) with \(R d\theta\). Also, since the arc is perpendicular to the plane containing point C, the magnetic field produced at point C will be directed towards the center of the circular arc. Thus, the expression for the magnetic field due to the circular arc becomes: \(d B_{arc} = \frac{\mu_{0} i}{4 \pi R} R d\theta \times \frac{1}{R^3}\). Now, integrate the expression above from \(0\) to \(\pi / 2\): \(B_{arc} = \int_{0}^{\pi / 2} d B_{arc} = \int_{0}^{\pi / 2} \frac{\mu_{0} i}{4 \pi R^2} R d\theta = \frac{\mu_{0} i}{4 \pi R} \int_{0}^{\pi / 2} R d\theta = \frac{\mu_{0} i}{4 \pi R} \left[\frac{\pi}{2}R\right]\). Thus, the magnetic field due to the circular arc is \(B_{arc} = \frac{\mu_{0} i}{8 R}\).

Determine the magnetic field due to the straight wire sections

Since the straight wire sections are symmetrical and mutually perpendicular, their magnetic field contributions are equal in magnitude but opposite in direction. This means that the net magnetic field due to their combined effect at point C is zero.

Determine the total magnetic field at point C

To find the total magnetic field at point C, sum up the contributions from the circular arc (\(B_{arc}\)) and the straight wire sections (which is zero): \(B_C = B_{arc} = \frac{\mu_{0} i}{8 R}\). Thus, the magnetic field at the center of the circular arc, point C, is given by the option (d) \(B=\frac{\mu_{0} i}{8 R}\).