Question

the figure shows four arrangements in which long parallel wires carry equal currents directly into or out of the page at the corners of identical squares. rank the arrangements according to the magnitude of the net magnetic field at the center of the square, greatest first.
(a)
(b)
(c)
(d)
○ a = b = c = d
○ c > d > a = b
○ c > d > a > b
○ a > b > c > d

Explanation:

Step1: Recall magnetic - field formula for a long - straight wire

The magnetic field due to a long - straight wire at a distance $r$ from it is given by $B=\frac{\mu_0I}{2\pi r}$. At the center of the square, for each wire, the distance from the wire to the center is the same. When currents are in the same direction, the magnetic fields due to the wires add up, and when currents are in opposite directions, the magnetic fields due to the wires subtract.

Step2: Analyze case (a)

In case (a), since all currents are either into or out of the page in the same sense, the magnetic fields due to the four wires cancel each other out at the center of the square. So, the net magnetic field $B_a = 0$.

Step3: Analyze case (b)

In case (b), two pairs of opposite wires have currents in opposite directions. The magnetic fields due to these pairs do not cancel completely. The net magnetic field is non - zero.

Step4: Analyze case (c)

In case (c), two adjacent wires have currents in one direction and the other two adjacent wires have currents in the opposite direction. The net magnetic field is non - zero and is larger than in case (b) because the magnetic fields due to the non - canceling components are more aligned.

Step5: Analyze case (d)

In case (d), similar to case (b), two pairs of opposite wires have currents in opposite directions, but the configuration results in a smaller net magnetic field compared to case (c).

The ranking of the magnitude of the net magnetic field at the center of the square from greatest to least is $c > d> a = b$.

Answer:

C. $c > d> a = b$