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 total magnetic field at the center of the circular loop is \(\frac{\mu_0 i}{2R}\), and its direction is perpendicular to and outward from the plane of the circular loop.

Step-by-step solution

Determine magnetic field due to parallel wires

We'll make use of Ampère's Law to determine the magnetic field produced by the parallel wires. For a long straight current-carrying wire, the magnetic field at a distance 'r' from the wire is given by the formula: \(B_{wire} = \frac{\mu_0 i}{2 \pi r}.\) Now, let's calculate the magnetic field produced by the two parallel wires at the center of the circular loop (Point 'O' in the diagram). We know that the distance between the parallel wires is 'D', and the radius of the circular loop is \(R = D/2\). Therefore, the distance of each wire from the center of the circular loop is \(r = R\). For wire 1, the magnetic field is: \(B_{1} = \frac{\mu_0 i}{2 \pi R}\), and for wire 2, the magnetic field is: \(B_{2} = \frac{\mu_0 i}{2 \pi R}\). The magnetic fields produced by both wires at point 'O' are equal in magnitude, but they act in opposite directions. Therefore, the net magnetic field at point O due to the parallel wires is the sum of both magnetic fields, i.e., \(B_{net} = B_{1} - B_{2} = \frac{\mu_0 i}{2 \pi R} - \frac{\mu_0 i}{2 \pi R} = 0\).

Determine magnetic field due to the circular loop

We'll now determine the magnetic field produced by the circular loop at its center (Point 'O' in the diagram). We'll again use Ampère's Law to determine this. The magnetic field produced by a current-carrying circular loop of radius 'R' at its center is given by the formula: \(B_{circle} = \frac{\mu_0 i R}{2(R^2)^{3/2}} = \frac{\mu_0 i}{2R}\).

Determine total magnetic field at the center of the loop

Since we already know the magnetic field contributions due to the parallel wires and the circular loop, the total magnetic field at the center of the loop is the sum of these magnetic fields. In this case, since the magnetic field due to the parallel wires is zero, the total magnetic field at the center of the loop is simply the magnetic field due to the circular loop. \(B_{total} = B_{net} + B_{circle} = 0 + \frac{\mu_0 i}{2R} = \frac{\mu_0 i}{2R}\). The direction of this magnetic field is perpendicular and outward from the plane of the circular loop since the current in the loop is counterclockwise. Therefore, the magnitude of the magnetic field at the center of the circular loop is \(\frac{\mu_0 i}{2R}\), and its direction is perpendicular to and outward from the plane of the circular loop.

Key concepts

Ampère's Law

Understanding Ampère's Law is crucial when delving into the principles of electromagnetism, especially regarding magnetic fields due to current flow.

Ampère's Law states that the integrated magnetic field around a closed path, often referred to as an Amperian loop, is directly proportional to the electric current flowing through the loop. In a more straightforward formula, it is denoted as \[ \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} \. \] Here, \( \vec{B} \) represents the magnetic field, \( d\vec{l} \) is an infinitesimal element of the loop path, and \( I_{enc} \) is the enclosed current by the loop. The constant \( \mu_0 \) is the permeability of free space, indicating how much resistance is encountered when forming a magnetic field in the vacuum.

Ampère's Law is of particular importance when the symmetry of the system allows us to apply it effectively, such as for long straight wires, solenoids, or toroids. In our exercise, it helps determine the magnetic field generated by both the current-carrying wires and the circular loop.

Magnetic Field due to Current-Carrying Wire

The magnetic field around a current-carrying wire is a tangible demonstration of electromagnetism in action. The formula derived from Ampère's Law for a long straight wire is \( B_{wire} = \frac{\mu_0 i}{2 \pi r} \.\) This indicates the magnetic field (\(B_{wire}\)) relies on the current (\(i\)) flowing through the wire and the distance (\(r\)) from it.

The idea presented here is that a magnetic field forms concentric circles around the wire, with the field strength diminishing as you move further away. It is critical to remember that the direction of the field is given by the right-hand rule, where a thumb pointing in the direction of the current flow indicates the magnetic field lines wrapping around the wire. In our example, we examined two parallel wires carrying equal but opposite currents, which results in mutually cancelled magnetic fields at the center of the loop between them. This cancellation is key to understanding the net magnetic field in systems with multiple current paths.

Magnetic Field due to Circular Loop

When a current flows through a circular loop, it creates a magnetic field distinct from that of a straight wire. For such a loop, the magnetic field at the centre can be computed using the formula \[ B_{circle} = \frac{\mu_0 i}{2R} \.\] The striking difference lies in the arrangement of the wire and how the current circulates.

This field (\(B_{circle}\)) at the center of the loop is also directed according to the right-hand rule, with your fingers following the current's direction and the thumb pointing towards the center, indicating the direction of the magnetic field.

In our exercise context, the circular loop encapsulates the parallel wires, contributing its magnetic field at the center. Even though the individual contributions from the parallel wires ultimately nullify each other, the loop's field persists, ensuring that the total magnetic field at the center is derived solely from the circular loop, guiding us to the final solution.