Calculating Magnetic Field at the Center of a Square Wire Loop

The problem asks for the magnetic field at the point of intersection of the diagonals of a square wire loop carrying a current I. This point is the geometric center of the square. A square wire loop consists of four straight current-carrying segments.

To find the total magnetic field at the center, we can use the principle of superposition. This means we calculate the magnetic field produced by each straight wire segment (side of the square) at the center and then add them vectorially.

Magnetic Field Due to a Single Straight Wire Segment

Consider one side of the square wire loop. Let the side length be L. The point of interest is the center of the square. The perpendicular distance 'r' from the center of the square to the midpoint of each side is half the side length, i.e., \(r = L/2\).

The magnetic field at a point due to a finite straight current-carrying wire segment is given by:

\( B = \frac{\mu_0 I}{4\pi r} (\sin \theta_1 + \sin \theta_2) \)

where:

• \(\mu_0\) is the permeability of free space.
• \(I\) is the current in the wire.
• \(r\) is the perpendicular distance from the point to the wire.
• \(\theta_1\) and \(\theta_2\) are the angles subtended by the lines joining the ends of the wire segment to the point, with the perpendicular line from the point to the wire.

For a side of the square of length L, the perpendicular distance from the center to the side is \(r = L/2\). The angles \(\theta_1\) and \(\theta_2\) for the center point relative to the ends of the side are the angles formed by the lines from the center to the corners with the perpendicular to the side. Since the center lies on the diagonal and the perpendicular bisects the side, these angles are 45 degrees (\(\pi/4\) radians).

So, \(\theta_1 = 45^\circ\) and \(\theta_2 = 45^\circ\).

Now, we can calculate the magnetic field due to one side:

\( B_{side} = \frac{\mu_0 I}{4\pi (L/2)} (\sin 45^\circ + \sin 45^\circ) \)

\( B_{side} = \frac{\mu_0 I}{2\pi L} (\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}) \)

\( B_{side} = \frac{\mu_0 I}{2\pi L} (\frac{2}{\sqrt{2}}) \)

\( B_{side} = \frac{\mu_0 I}{2\pi L} (\sqrt{2}) \)

\( B_{side} = \frac{\sqrt{2} \mu_0 I}{2\pi L} \)

Direction of Magnetic Field

Using the right-hand rule, if we assume the current is flowing in a specific direction (e.g., clockwise or counter-clockwise) around the loop, the magnetic field produced by each straight segment at the center will point in the same direction (either into the page or out of the page, depending on the current direction).

For a loop in the xy-plane, if the current is clockwise, the field at the center is into the plane (negative z direction). If the current is counter-clockwise, the field is out of the plane (positive z direction). Since all four sides contribute a field in the same direction at the center, we can simply add the magnitudes.

Total Magnetic Field at the Center

The total magnetic field at the center of the square loop is the sum of the magnetic fields due to all four sides:

\( B_{total} = B_{side1} + B_{side2} + B_{side3} + B_{side4} \)

Since the geometry and current are the same for all sides, \(B_{side1} = B_{side2} = B_{side3} = B_{side4} = B_{side}\).

\( B_{total} = 4 \times B_{side} \)

\( B_{total} = 4 \times \frac{\sqrt{2} \mu_0 I}{2\pi L} \)

\( B_{total} = \frac{4\sqrt{2} \mu_0 I}{2\pi L} \)

\( B_{total} = \frac{2\sqrt{2} \mu_0 I}{\pi L} \)

This calculated magnetic field at the point of intersection of diagonals matches one of the given options.

Summary of Calculation Steps

1. Identify the geometry and the point of interest (center of the square loop).
2. Recognize the loop as four straight wire segments.
3. Use the formula for the magnetic field due to a finite straight wire segment.
4. Determine the perpendicular distance from the center to a side (\(L/2\)).
5. Determine the angles subtended by the ends of the segment at the center (\(45^\circ\) or \(\pi/4\)).
6. Calculate the magnetic field magnitude due to one side.
7. Determine the direction of the magnetic field from each side at the center.
8. Apply superposition to find the total magnetic field by summing the contributions from all four sides.

Revision Table: Magnetic Field Calculations

Additional Information: Magnetic Fields from Current Loops

Understanding magnetic fields from current loops is fundamental in electromagnetism. Different shapes of loops produce different magnetic field patterns and magnitudes at specific points.

• Circular Loop: The magnetic field at the center of a circular loop of radius R carrying current I is given by \( B_{center} = \frac{\mu_0 I}{2R} \).
• Solenoid: A long coil of wire (solenoid) produces a nearly uniform magnetic field inside it, given by \( B = \mu_0 n I \), where n is the number of turns per unit length.
• Toroid: A solenoid bent into a circle forms a toroid. The magnetic field inside the toroid is confined and varies with radial distance.

The calculation for the square loop demonstrates how to apply the formula for a straight wire segment and the superposition principle to more complex shapes formed by straight segments.