Problem 3 - Fundamentals of Physics

Question

SSM At a certain location in the Philippines, Earth's magnetic field of \(39 \mu \mathrm{T}\) is horizontal and directed due north. Suppose the net field is zero exactly \(8.0 \mathrm{~cm}\) above a long, straight, horizontal wire that carries a constant current. What are the (a) magnitude and (b) direction of the current?

Step-by-Step Solution

Answer

The current is approximately 15.6 A, with the direction such that its magnetic field opposes Earth's field, pointing southward.

Step 1: Understanding the Problem

We are given that Earth's magnetic field is horizontal and directed due north with a magnitude of \(39 \mu\mathrm{T}\). We want to find the current in a wire such that 8 cm above it, the net magnetic field is zero.

Step 2: Set Up Magnetic Field Equation

The magnetic field \( B \) produced by a long, straight wire at a distance \( r \) from it is given by the formula \( B = \frac{\mu_0 I}{2\pi r} \), where \( \mu_0 = 4\pi \times 10^{-7} \frac{\mathrm{T}\cdot\mathrm{m}}{\mathrm{A}} \) is the permeability of free space, and \( I \) is the current. We need the wire's magnetic field to balance Earth's magnetic field.

Step 3: Calculate the Required Current

Since the net magnetic field is zero, the wire’s field must be equal in magnitude but opposite in direction to Earth's magnetic field. Set \( B = 39 \mu\mathrm{T} = 39 \times 10^{-6} \mathrm{T} \). Plug this into \( B = \frac{\mu_0 I}{2\pi r} \), replace \( r = 0.08 \) m, and solve for \( I \).

Step 4: Solve for Current

Rearrange the equation to find the current: \( I = \frac{B \cdot 2\pi r}{\mu_0} = \frac{39 \times 10^{-6} \times 2\pi \times 0.08}{4\pi \times 10^{-7}} \). Simplifying, \( I = \frac{39 \times 10^{-6} \times 0.08}{2 \times 10^{-7}} \).

Step 5: Calculate and Interpret Result

Calculate \[ I \approx \frac{39 \cdot 0.08}{2} \times 10 = 15.6 \mathrm{A} \]. The current required to counterbalance Earth's magnetic field is approximately 15.6 A. The direction must be such that it produces a magnetic field southward to cancel the Earth's northward field.

Key Concepts

Ampere's Law

Ampere's Law is a fundamental equation in electromagnetism that relates the magnetic field around a closed loop to the electric current passing through that loop. This law is expressed as: \[ \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} \]Where:

• \( \oint \vec{B} \cdot d\vec{l} \) is the line integral of the magnetic field \( \vec{B} \) around the closed path.
• \( \mu_0 \) is the permeability of free space, approximately equal to \( 4\pi \times 10^{-7} \frac{\mathrm{T}\cdot\mathrm{m}}{\mathrm{A}} \).
• \( I_{enc} \) is the current enclosed by the path.

Ampere's Law is particularly useful for calculating the magnetic field created by current-carrying conductors of simple geometries, such as the straight wire in our exercise. For a long, straight wire, the magnetic field strength decreases as the distance from the wire increases, described by the formula:\[ B = \frac{\mu_0 I}{2\pi r} \]For our particular exercise, this formula is used to determine the current required in the wire to match and counteract Earth's magnetic field at a certain distance. By setting this magnetic field equal to Earth's field (in magnitude), we can solve for the current \( I \).

Earth's Magnetic Field

Earth's magnetic field acts much like a giant bar magnet, with a north and south pole. It is crucial for navigation and has an average strength of roughly 25 to 65 microteslas (\(\mu\text{T}\)) depending on location. At the location mentioned in the exercise, the field is 39 \(\mu \text{T}\) and points horizontally northward. This natural magnetic field originates from the motion of molten iron in Earth's outer core. It extends from the earth's interior outward into space, forming the magnetosphere, which protects us from charged particles. In comparison to man-made fields, Earth's magnetic field is relatively weak, but it has a major influence over a large area.In the exercise, this field must be counteracted by the artificial magnetic field created by the current in a wire. To achieve a net magnetic field of zero, the wire's field must be equal in magnitude but opposite in direction at a specific point above the wire. This concept of balancing out fields is crucial to solving the exercise.

Current Direction

The direction of the current is crucial in determining the direction of the produced magnetic field. According to the right-hand rule, if you point your thumb in the direction of conventional current (positive to negative), your fingers will curl in the direction of the magnetic field surrounding the wire. In our exercise, Earth's magnetic field is directed northward; hence, to have a net field of zero, the wire's field must be directed southward at the point of interest. This opposite direction allows the magnetic influences to counteract one another. Calculating the needed direction:

• Using your right hand, position your thumb to point in the direction that would generate a southward field above the wire.
• The fingers should curl such that they demonstrate the circular path of the magnetic field lines produced by the wire.

Determining this direction ensures that the calculation of the current's magnitude and the solution's implementation achieves the desired nullification of Earth's field at the given location.