Question

Parallel wires, a distance \(D\) apart, carry a current, \(i\), in opposite directions as shown in the figure. A circular loop, of radius \(R=D / 2,\) has the same current flowing in a counterclockwise direction. Determine the magnitude and the direction of the magnetic field from the loop and the parallel wires at the center of the loop, as a function of \(i\) and \(R\).

Short answer

Answer: The magnitude of the magnetic field at the center of the circular loop is \(\dfrac{i\mu_0}{8\sqrt{2}R}\), and the direction of the magnetic field is into the plane.

Step-by-step solution

Determine the magnetic field at the center from each wire individually

(Using Ampere's Law, which states that the closed line integral of the magnetic field, \(\oint B \cdot dl= \mu_0 I_{enc}\), where \(I_{enc}\) is the enclosed current). We can use Ampere's Law to find the magnetic field \(B_w\) generated by the parallel wires: $$ B_w = \dfrac{\mu_0 i}{2\pi R}$$ For the magnetic field generated by the circular loop \(B_l\), we can use the formula: $$ B_l = \dfrac{\mu_0i R}{2 (R^2+z^2)^{3/2}}$$ where \(z\) is the distance from the loop plane to the center of the loop (in this case \(z = R\)).

Calculate the total magnetic field at the center of the loop

(Summing the magnetic fields of the parallel wires and the circular loop) The magnetic fields from the parallel wires are anti-parallel to each other and hence, they cancel each other out. We are left with the magnetic field contribution from the circular loop. Substituting the values of \(R\) and \(z\), we get: $$ B_l = \dfrac{\mu_0i R}{2 (R^2+R^2)^{3/2}}$$ Simplifying the expression: $$ B_l = \dfrac{\mu_0i R}{2 (2R^2)^{3/2}}$$ After simplification, we get: $$ B_l = \dfrac{i\mu_0}{8\sqrt{2}R}$$

Express the magnitude and direction of the magnetic field

(Using the final expression for the magnetic field of the circular loop) The magnitude of the magnetic field at the center of the loop is: $$\lvert B_l \rvert = \dfrac{i\mu_0}{8\sqrt{2}R} $$ The direction of the magnetic field at the center of the loop is due to the counterclockwise current in the circular loop. This will create a magnetic field that is directed into the plane (using the right-hand rule for magnetic fields). In conclusion, the magnitude of the magnetic field at the center of the circular loop is \(\dfrac{i\mu_0}{8\sqrt{2}R}\), and the direction of the magnetic field is into the plane.