Problem 52 - Physics

Question

A very long, straight wire carries a current of \(0.12 \mathrm{~A}\). This wire is tangent to a singleturn, circular wire loop that also carries a current. The directions of the currents are such that the net magnetic field at the center of the loop is zero. Both wires are insulated and have diameters that can be neglected. How much current is there in the loop?

Step-by-Step Solution

Answer

The loop current is approximately 0.038 A.

Step 1: Understand the condition for zero net magnetic field

The net magnetic field at the center of the circular loop is the sum of the magnetic fields due to the current in the loop and the long wire. Since they are equal and opposite, they cancel each other out.

Step 2: Express the magnetic field due to the straight wire

The magnetic field at a distance \( r \) from a long straight wire carrying current \( I \) is given by \( B = \frac{\mu_0 I}{2 \pi r} \), where \( \mu_0 \) is the permeability of free space.

Step 3: Express the magnetic field due to the loop

The magnetic field at the center of a circular loop carrying a current \( I \) with radius \( R \) is given by \( B = \frac{\mu_0 I}{2R} \).

Step 4: Set the two magnetic fields to be equal

Set the magnetic field due to the straight wire equal to the magnetic field due to the loop to find the relation: \( \frac{\mu_0 I_{\text{wire}}}{2 \pi r} = \frac{\mu_0 I_{\text{loop}}}{2R} \).

Step 5: Solve for the loop current

Rearrange the equation from Step 4 to solve for the loop current \( I_{\text{loop}} = \frac{R}{\pi r} I_{\text{wire}} \). Since \( R = r \), it simplifies to \( I_{\text{loop}} = \frac{1}{\pi} I_{\text{wire}} \). Substitute \( I_{\text{wire}} = 0.12 \) A to obtain \( I_{\text{loop}} = \frac{0.12}{\pi} \) A.

Key Concepts

Ampere's Law

Ampere's Law is a fundamental principle used to calculate magnetic fields, especially for configurations involving symmetries like long straight wires and loops. It states that the magnetic field created by an electric current is proportional to the current multiplied by the length of the path through which the field circulates, and inversely proportional to the distance from the current-carrying conductor. Ampere’s Law can be expressed mathematically with the line integral of the magnetic field along a closed path: \[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I, \]where \(\mathbf{B}\) is the magnetic field vector, \(d\mathbf{l}\) is a small segment of the path, \(I\) is the total current enclosed by the path, and \(\mu_0\) is the permeability of free space. This principle is crucial when determining how the magnetic fields generated by different currents interact, such as in the case of a straight wire and a circular loop.

Current in a Loop

When a current flows in a circular loop, it generates a magnetic field. At the center of a loop of radius \( R \), the magnetic field can be described by the formula: \[ B = \frac{\mu_0 I}{2R}, \]where \( I \) is the current flowing through the loop. The direction of this field follows the right-hand rule: curl your fingers in the direction of the current, and your thumb points in the direction of the field. In scenarios where a loop and a straight wire are involved, understanding these fields is essential. To achieve magnetic field cancellation, the loop must carry a specific current that balances out the field produced by the wire. This cancels both fields at the loop's center, leading to a net magnetic field of zero. Current behavior in loops helps understand devices like transformers and electromagnets, where loop arrangements are common.

Magnetic Field Cancellation

Magnetic field cancellation occurs when two magnetic fields are equal in magnitude but opposed in direction. In our example, the magnetic field created by the long straight wire carrying a current of 0.12 A is cancelled by the field from the circular loop's current.When the fields from the wire and the loop add up to zero at the center of the loop, this condition is attained by setting the fields equal in magnitude but opposite in direction:\[ \frac{\mu_0 I_{\text{wire}}}{2 \pi r} = \frac{\mu_0 I_{\text{loop}}}{2R}. \]Once set appropriately, it allows us to find the exact current needed in the loop to achieve nullification, not only balancing the magnetic forces but also providing insight into real-world applications, like in token ring network topology or magnetic shielding.

Permeability of Free Space

The permeability of free space, represented as \( \mu_0 \), is a constant that characterizes the ability of the vacuum of space to support a magnetic field. It is crucial in electromagnetism and appears in many key equations, including those used to determine magnetic field strength around conductors and loops. Its value is approximately \( 4\pi \times 10^{-7} \text{ T} \cdot \text{m/A} \). This fundamental constant bridges the relationship between electric currents and magnetic fields, governing how fields spread and interact through space.Understanding \(\mu_0\) aids in grasping how magnetic fields function not only in vacuums but in materials modulating them, relevant for designing electromagnets, inductors, and other magnetic devices.