Question

the figure shows three circuits consisting of straight radial lengths and concentric circular arcs (either half - or quarter - circles of radii r, 2r, and 3r). the circuits carry the same current. rank them according to the magnitude of the magnetic field produced at the center of curvature (the dot), greatest first. (a) (b) (c) o a = b = c o b > a > c o c > b > a o c > a > b o a > b > c o a > c > b o b > c > a

Explanation:

Step1: Recall magnetic - field formula for circular arc

The magnetic field at the center of a circular arc of radius \(r\) carrying current \(I\) is given by \(B=\frac{\mu_0I\theta}{4\pi r}\), where \(\theta\) is the angle subtended by the arc at the center. For the given semi - circular arcs, \(\theta=\pi\). The straight radial segments do not contribute to the magnetic field at the center (since the current is along the line joining the center and the element of the straight segment, so \(\vec{dl}\times\vec{r} = 0\)).

Step2: Analyze the effect of radius on magnetic field

Since \(B=\frac{\mu_0I\pi}{4\pi r}=\frac{\mu_0I}{4r}\), the magnetic field \(B\) is inversely proportional to the radius \(r\) of the circular arc. The smaller the radius, the larger the magnetic field.

Step3: Compare the radii of the given arcs

Let the radii of the arcs in (a), (b), and (c) be \(r_a\), \(r_b\), and \(r_c\) respectively. If we assume \(r_a B_b>B_c\).

Answer:

C. \(a > b > c\)