Question

A straight, 2.5-m wire carries a typical household current of 1.5 A (in one direction) at a location where the earth's magnetic field is 0.55 gauss from south to north. Find the magnitude and direction of the force that our planet's magnetic field exerts on this wire if it is oriented so that the current in it is running (a) from west to east, (b) vertically upward, (c) from north to south. (d) Is the magnetic force ever large enough to cause significant effects under normal household conditions?

Short answer

Forces: (a) upward 2.0625e-4 N, (b) east 2.0625e-4 N, (c) 0 N; insignificant forces for households.

Step-by-step solution

Convert Magnetic Field Units

The magnetic field strength is given in gauss. To use it in the formula, convert it to tesla. We know that 1 gauss equals \(10^{-4}\) tesla, so 0.55 gauss would be \(0.55 \times 10^{-4} = 5.5 \times 10^{-5}\) tesla.

Use the Lorentz Force Formula

The force on a current flowing in a wire in a magnetic field is given by the formula \( F = I L B \sin \theta \), where \( I \) is the current in amperes, \( L \) is the length of the wire in meters, \( B \) is the magnetic field in tesla, and \( \theta \) is the angle between the direction of the current and the direction of the magnetic field.

Calculate Force for Current West to East

For (a), the current is from west to east, and the earth's magnetic field is from south to north. Thus, \( \theta = 90^\circ \), and \( \sin 90^\circ = 1 \).Substitute into the formula:\[ F = 1.5 \times 2.5 \times 5.5 \times 10^{-5} \times 1 \]\[ F = 2.0625 \times 10^{-4} \text{ N} \]The force direction is using the right-hand rule: upward.

Calculate Force for Vertically Upward Current

For (b), the current is vertical, and the earth's magnetic field is horizontal, meaning \( \theta = 90^\circ \) and \( \sin 90^\circ = 1 \).Substitute into the formula:\[ F = 1.5 \times 2.5 \times 5.5 \times 10^{-5} \times 1 \]\[ F = 2.0625 \times 10^{-4} \text{ N} \]The force direction is determined by the right-hand rule: horizontal, from west to east.

Calculate Force for Current North to South

For (c), the current is aligned with the magnetic field, meaning \( \theta = 0^\circ \), so \( \sin 0^\circ = 0 \).Substitute into the formula:\[ F = 1.5 \times 2.5 \times 5.5 \times 10^{-5} \times 0 \]\[ F = 0 \text{ N} \]There is no force when the current is parallel to the magnetic field.

Evaluate Significance of Magnetic Force

For (d), consider that the forces calculated are in the order of \(10^{-4}\) newtons, which is extremely small. Such a force is not significant and unlikely to influence household appliances or cause visible effects.

Key concepts

Magnetic Field

A magnetic field is an invisible force field created by moving electric charges, which can interact with similar fields and produce forces. In this exercise, we are concerned with the Earth's magnetic field, which extends from south to north. The strength of the magnetic field is often measured in units called gauss or tesla.
For this problem, it's given as 0.55 gauss. To make calculations easier, we convert gauss to tesla, where 1 gauss equals \(10^{-4}\) tesla. Therefore, 0.55 gauss is equivalent to \(5.5 \times 10^{-5}\) tesla, which gives us the strength of the Earth's magnetic field in a format that's ready for calculations.
These magnetic fields are crucial in many physical applications, such as in electric motors, generators, and, as illustrated here, the interaction with current-carrying wires.

Current Wire Interaction

When current flows through a wire, it creates its own magnetic field around it. This magnetic field interacts with any other magnetic fields present, such as that of the Earth. The result of this interaction is a magnetic force being applied to the wire.
The strength and direction of this force depend on several factors, including the current's magnitude, the length of the wire, and the angle between the current direction and the magnetic field direction. The current in this scenario is given as 1.5 A and the wire is 2.5 meters long.
This interaction showcases a fundamental concept in physics, demonstrating how electric and magnetic fields influence each other. It forms the basis for the operation of various devices and technologies, illustrating the power of electromagnetic interactions.

Right-Hand Rule

The right-hand rule is a helpful tool used to determine the direction of the magnetic force acting on a current-carrying wire. When your right hand's thumb points in the current's direction and your fingers point along the magnetic field lines, your palm shows the direction of the force.
In this particular exercise, you apply the right-hand rule to determine how the earth's magnetic field influences the current. For instance, when current flows from west to east and Earth's magnetic field travels south to north, the force is directed upwards.
Understanding and applying this rule is essential for accurately predicting and analyzing the behavior of systems where magnetic fields and currents interact. It's a critical visualization technique for students and professionals alike.

Magnetic Force Calculation

Calculating the magnetic force on a wire involves using the Lorentz force formula: \( F = I L B \sin \theta \). Here, \( I \) is the current, \( L \) is the wire length, \( B \) is the magnetic field strength, and \( \theta \) is the angle between the current and the magnetic field.
For example, when current flows from west to east, and Earth's magnetic field goes from south to north, the angle \( \theta \) is \( 90^\circ \). Thus, \( \sin 90^\circ = 1 \), and our formula simplifies to \( F = 1.5 \times 2.5 \times 5.5 \times 10^{-5} \), resulting in a force of \( 2.0625 \times 10^{-4} \) N exerted upwards.
If the current aligns with the magnetic field, \( \theta = 0^\circ \) and \( \sin 0^\circ = 0 \), leading to zero force. These calculations elucidate how magnetic forces vary with current orientation and reinforce the importance of understanding vector directions in electromagnetic contexts.