Problem 57 - College Physics

Question

A circular metal loop is 22 \(\mathrm{cm}\) in diameter. (a) How large a current must flow through this metal so that the magnetic field at its center is equal to the earth's magnetic field of \(0.50 \times\) \(10^{-4} \mathrm{T}\) (b) Show how the loop should be oriented so that it can cancel the earth's magnetic field at its center.

Step-by-Step Solution

Answer

(a) The current must be approximately 87.2 A. (b) The loop should be oriented to produce a magnetic field opposite to Earth's field.

Step 1: Understand the Problem

We are asked to find the current needed in a circular loop to create a magnetic field at its center that equals the Earth's magnetic field and the loop's orientation to cancel the Earth's magnetic field.

Step 2: Recall the Magnetic Field Formula

The magnetic field at the center of a circular loop with current is given by the formula: \( B = \frac{\mu_0 I}{2R} \), where \( B \) is the magnetic field, \( \mu_0 = 4\pi \times 10^{-7} \ \mathrm{T \, m/A} \) is the permeability of free space, \( I \) is the current, and \( R \) is the radius of the loop.

Step 3: Convert Diameter to Radius

The diameter of the loop is given as 22 cm. Therefore, the radius \( R \) is \( \frac{22}{2} = 11 \ \mathrm{cm} = 0.11 \ \mathrm{m} \).

Step 4: Calculate the Required Current

Using the formula \( B = \frac{\mu_0 I}{2R} \) and rearranging for \( I \), we have \( I = \frac{2BR}{\mu_0} \). Substitute \( B = 0.50 \times 10^{-4} \ \mathrm{T} \), \( R = 0.11 \ \mathrm{m} \), and \( \mu_0 = 4\pi \times 10^{-7} \ \mathrm{T \, m/A} \):\[ I = \frac{2 \times 0.50 \times 10^{-4} \times 0.11}{4\pi \times 10^{-7}} \approx 87.2 \ \mathrm{A} \].

Step 5: Determine Loop Orientation for Cancellation

To cancel the Earth's magnetic field at the center, the magnetic field produced by the loop must be equal in magnitude but opposite in direction to the Earth's magnetic field. This means the magnetic field vector produced by the loop should be oriented opposite to Earth's field vector.

Key Concepts

Circular Loop

A circular loop is essentially a closed loop which can carry electric current. These loops are crucial in physics, as they help demonstrate the principles of magnetic fields created by currents.
When a current passes through a circular loop, it generates a magnetic field, predominantly concentrated at the center of the loop. This field can be significant and measurable.
These loops are often made of conductive material, such as metal, to allow the electrons to flow smoothly, producing a steady current.

• Diameter is a measure from one edge of the loop to the other.
• The radius, which is half the diameter, is important for calculations as it represents the distance from the center to any point on the loop.

By understanding diameter and radius, we can move to more complex calculations involving magnetic fields in such loops.

Current Calculation

Calculating the current in a circular loop to produce a desired magnetic field involves using the magnetic field formula. The task is usually to find the current necessary to achieve a certain magnetic field strength, such as the Earth's magnetic field in this exercise.
To calculate the required current, you use the formula for the magnetic field generated by the loop: \[ I = \frac{2BR}{\mu_0} \] where

• \( I \) is the current,
• \( B \) is the desired magnetic field strength,
• \( R \) is the radius of the loop.

The symbol \( \mu_0 \) is the permeability of free space, a constant which helps relate the magnetic field to the current responsible for it. By plugging in the known values, you can solve for the unknown current. Here, for a given magnetic field \( B \), and a loop with radius \( R \), this formula gives the current \( I \) needed to generate that magnetic field.

Earth's Magnetic Field

Earth's magnetic field is a vital component influencing navigation, atmospheric conditions, and greater environmental systems. It resembles the field of a giant bar magnet tilted about 11 degrees from the Earth’s rotational axis.
On the surface, its strength varies from around 25 to 65 microteslas, influencing compass needles around the world. It acts as a shield against solar wind, a stream of charged particles emanating from the sun.

• Earth's magnetic field affects devices by aligning their magnetic elements.
• It helps migratory animals navigate long distances.

Understanding Earth's magnetic field helps us comprehend phenomena like auroras and is crucial in the study of geology and space physics. In situations involving artificial magnetic fields, like the circular loop exercise, understanding Earth's field provides context for why certain orientations and strength adjustments are necessary.

Magnetic Field Formula

The magnetic field formula is an essential element in electromagnetism. It calculates the magnetic field strength that a current produces in a loop. The formula for the magnetic field at the center of a circular loop is as follows: \[ B = \frac{\mu_0 I}{2R} \]This formula calculates \( B \), the magnetic field strength:

• \( \mu_0 \), the permeability of free space, is a constant with a value of \( 4\pi \times 10^{-7} \ \mathrm{T \ m/A} \)
• \( I \), the current through the loop, determines how strong the magnetic field will be
• \( R \), the radius of the loop, impacts the field's distribution

Using this formula, you can solve for any of these variables, given the others. It aids in determining the behavior of magnetic fields in response to current flow, crucial for designing systems that utilize electromagnetism like motors, generators, and magnetic field cancellation systems as explored in the given problem.