Question

An electric current i = 0.85 A is flowing in a long wire. Consider a rectangular area with one side parallel to the wire and at a distance c = 0.024 m away from the wire. Let the dimensions of the rectangle be a = 0.018 m and b = 0.074 m. What is the magnetic field at the center of the rectangle?

Answer 1

The magnetic field at the center of the rectangle due to the current flowing in the wire is
`\( B = 2.04 * 10^(-6) \, T \).`

Step-by-step explanation:

The magnetic field
`(\( B \))` at the center of the rectangle can be determined using Ampere's Law. For a long straight wire, the formula is given by
`\( B = (\mu_0 \cdot i)/(2\pi \cdot r) \), where \( \mu_0 \)` is the permeability of free space,
`\( i \)` is the current, and
`\( r \)` is the distance from the wire. In this case, the wire creates a magnetic field that circulates around it. The rectangle, being parallel to the wire, intercepts some of these field lines.

Now, consider a rectangular loop formed by the sides of the rectangle perpendicular to the wire. Applying Ampere's Law to this loop, the formula becomes
`\( B = (\mu_0 \cdot i)/(2) \cdot \ln\left((b)/(a)\right) \).` Here,
`\( a \) and \( b \)` are the dimensions of the rectangle. Plugging in the given values, we get
`\( B = (4\pi * 10^(-7) \cdot 0.85)/(2) \cdot \ln\left((0.074)/(0.018)\right) \).`

Calculating this gives the final answer of
`\( B = 2.04 * 10^(-6) \, T \).`Therefore, the magnetic field at the center of the rectangle is
`\( 2.04 * 10^(-6) \, T \).`