Problem 39 - University Physics with Modern Physics

Question

Two concentric circular loops of wire lie on a tabletop, one inside the other. The inner wire has a diameter of 20.0 cm and carries a clockwise current of 12.0 A, as viewed from above, and the outer wire has a diameter of 30.0 cm. What must be the magnitude and direction (as viewed from above) of the current in the outer wire so that the net magnetic field due to this combination of wires is zero at the common center of the wires?

Step-by-Step Solution

Answer

The outer loop must carry 18.0 A clockwise for the net magnetic field to be zero.

Step 1: Understand Magnetic Field Due to a Circular Loop

A circular current-carrying loop generates a magnetic field at its center, which can be given by the formula: \[ B = \frac{\mu_0 I}{2r} \]where \( B \) is the magnetic field, \( \mu_0 \) is the permeability of free space \( (4\pi \times 10^{-7} \ \text{T}\cdot\text{m/A}) \), \( I \) is the current, and \( r \) is the radius of the loop.

Step 2: Calculate Magnetic Field of Inner Loop

Given the diameter of the inner loop as 20.0 cm, the radius \( r_1 \) is 10.0 cm or 0.1 m. The current \( I_1 \) is 12.0 A.Calculate the magnetic field at the center due to the inner loop:\[ B_1 = \frac{4\pi \times 10^{-7} \times 12}{2 \times 0.1} = \frac{4.8\pi \times 10^{-6}}{0.2} = 24\pi \times 10^{-6} \ \text{T} \approx 75.4 \times 10^{-6} \ \text{T} \]

Step 3: Find Required Magnetic Field of Outer Loop

To make the net magnetic field zero at the center, the magnetic field due to the outer loop \( B_2 \) must have the same magnitude as \( B_1 \) but in the opposite direction. Therefore, \( B_2 = -B_1 \).

Step 4: Calculate Required Current in Outer Loop

The diameter of the outer loop is 30.0 cm, so the radius \( r_2 \) is 15.0 cm or 0.15 m.Use the magnetic field formula in Step 1 with \( B_2 = -75.4 \times 10^{-6} \ \text{T} \) to find \( I_2 \):\[ 75.4 \times 10^{-6} = \frac{4\pi \times 10^{-7} \times I_2}{2 \times 0.15} \]\[ I_2 = \frac{75.4 \times 10^{-6} \times 0.3}{4\pi \times 10^{-7}} = \frac{22.62 \times 10^{-6}}{1.25664 \times 10^{-6}} \approx 18.0 \ \text{A} \]The outer loop must carry a current of 18.0 A.

Step 5: Determine Direction of Magnetic Field in Outer Loop

Since the inner current is clockwise and creates a field pointing into the plane, the current in the outer loop must also be clockwise to create a field that points out of the plane, thus canceling \( B_1 \).

Key Concepts

circular current loop

Understanding the concept of a circular current loop is crucial for analyzing magnetic fields. A circular current loop consists of a wire formed into a circular shape, carrying an electric current around it. This configuration generates a magnetic field that is strongest at the center of the loop.

The direction of the magnetic field produced is determined by the "right-hand rule." You can point the thumb of your right hand in the direction of the conventional current flow (from positive to negative), and your fingers will curl in the direction of the induced magnetic field. This simple hand gesture helps predict the orientation of the magnetic field lines.

• The stronger the current, the stronger the magnetic field generated.
• The radius of the loop is inversely proportional to the strength of the magnetic field at its center.

These simple rules allow us to predict how the magnetic field behavior changes with different loop sizes and current strengths.

current direction

The direction of the current in loops is pivotal to understanding how magnetic fields act. In this particular problem, the inner and outer loops both carry currents that affect their respective magnetic fields.

Current direction is often described as either clockwise or counterclockwise when the loop is viewed from above. The direction will determine whether the magnetic field produced by the loop points into or out of the plane of the loop.

• A clockwise current will generally produce a magnetic field pointing into the plane of the loop.
• Conversely, a counterclockwise current produces a magnetic field pointing out of the plane.

In this problem, since the magnetic field produced by the inner loop points into the plane, the outer loop current must also be clockwise to cancel it by pointing outwards.

magnetic field cancellation

Magnetic field cancellation is a phenomenon where the net magnetic field at a point is zero, due to equal and opposite fields from different sources. In the given exercise, this principle is employed by adjusting the current and direction in the outer loop to offset the magnetic field produced by the inner loop at the center.

To achieve cancellation, two primary conditions must be met:

• The magnitude of the magnetic fields from the loop must be equal but opposite in direction.
• The current direction must create a field that directly opposes the other field.

In the scenario of two concentric circular loops, this involves manipulating the magnitude and direction of the current in the outer loop. Using equations for magnetic fields and solving for the conditions where the net magnetic contribution becomes zero is essential in engineering applications like MRI machines, which use similar principles to focus and manage magnetic fields.

magnetic permeability

Magnetic permeability (\( \mu_0 \)) is a constant that plays a significant role in calculating magnetic fields. It is a measure of how medium responds to the presence of a magnetic field, fundamentally determining how much magnetic field is produced by the current in the loop.

The magnetic field at the center of a circular current loop can be calculated using the formula:\[ B = \frac{\mu_0 I}{2r} \]In this context, \( \mu_0 \) is the magnetic permeability of free space and is equal to \(4\pi \times 10^{-7} \ \text{T}\cdot\text{m/A}\). This constant applies universally in free space and allows us to predict the resulting magnetic field for any given current and radius of the loop.

• It is an intrinsic property of the medium surrounding the loop.
• Higher magnetic permeability implies a stronger magnetic field for the same amount of current.

This concept is fundamental in understanding how electromagnets work and plays a critical role in designing devices and circuits that rely on magnetic fields.