Problem 36 - University Physics with Modern Physics

Question

A straight, \(2.5-\mathrm{m}\) wire carries a typical household current of 1.5 \(\mathrm{A}\) (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 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 undernormal household conditions?

Step-by-Step Solution

Answer

The magnetic forces are downward (west-east current), eastward (vertical current), and zero (north-south current) with a maximum of \(2.0625 \times 10^{-4} \, \text{N} \). These forces are too small to affect household conditions.

Step 1: Identify Relevant Formula

To solve this problem, we start by recalling the formula for the magnetic force on a current-carrying wire, which is given by \[ F = I \cdot L \cdot B \cdot \sin(\theta) \]where \( F \) is the force, \( I \) is the current, \( L \) is the length of the wire, \( B \) is the magnetic field, and \( \theta \) is the angle between the direction of the current and the magnetic field.

Step 2: Change of Units for the Magnetic Field

The magnetic field needs to be converted from gauss to tesla. 1 gauss = \( 10^{-4} \) tesla, so 0.55 gauss = 0.000055 tesla.

Step 3: Calculate Force for Current from West to East

For part (a), when the current flows from west to east, it is perpendicular to the magnetic field (which is from south to north), making \( \theta = 90^\circ \). This makes \( \sin(90^\circ) = 1 \).The force can be calculated as:\[ F = 1.5 \times 2.5 \times 0.000055 \times 1 = 2.0625 \times 10^{-4} \, \text{N} \]The direction of the force can be determined by the right-hand rule and will point downward.

Step 4: Calculate Force for Current Vertically Upward

For part (b), when the current flows vertically upward, the angle \( \theta \) is \( 90^\circ \) because it is perpendicular to the horizontal field.The force will be:\[ F = 1.5 \times 2.5 \times 0.000055 \times 1 = 2.0625 \times 10^{-4} \, \text{N} \]Again, use the right-hand rule to find the direction. The force is directed eastward.

Step 5: Calculate Force for Current from North to South

For part (c), when the current flows from north to south, it is parallel to the magnetic field. Thus, \( \theta = 0^\circ \), making \( \sin(0^\circ) = 0 \).The force will be:\[ F = 1.5 \times 2.5 \times 0.000055 \times 0 = 0 \, \text{N} \]There is no force exerted in this orientation.

Step 6: Assess the Magnitude of the Force for Practical Effects

For part (d), we assess whether the magnetic force is significant.The calculated maximum force of \( 2.0625 \times 10^{-4} \, \text{N} \) is extremely small compared to household objects and cannot cause significant effects on daily household operations.

Key Concepts

Right-Hand Rule

Have you ever wondered how to determine the direction of a magnetic force on a current-carrying wire? The right-hand rule is here to help! It's a straightforward method used to find the direction of these forces. When applying this rule, point your right thumb in the direction of the current. Then, let your fingers unfurl in the direction of the magnetic field. Your palm will point in the direction of the force exerted on the wire.

In the context of our exercise, if the current flow is from west to east and the magnetic field is from south to north, by using the right-hand rule, you would find that the palm (force direction) points downward.

• Right thumb: direction of current
• Fingers: direction of magnetic field
• Palm: direction of force

This rule is crucial for visualizing and accurately determining the result of the interaction between current and magnetic fields.

Magnetic Field Conversion

Before diving into calculations involving magnetic force, it is crucial to ensure all units are compatible. In the exercise, the Earth's magnetic field is given in gauss, a unit not commonly used in physics calculations involving current-carrying wires.

The conversion of magnetic field strength from gauss to tesla is simple. Remember, 1 gauss equals \(10^{-4}\) tesla. So, to convert 0.55 gauss into tesla, multiply by \(10^{-4}\), giving you 0.000055 tesla.

• Ensure all measurements are in compatible units.
• 1 gauss = \(10^{-4}\) tesla
• Conversion: 0.55 gauss = 0.000055 tesla

Doing this conversion ensures that when you calculate the magnetic force using the formula \( F = I \cdot L \cdot B \cdot \sin(\theta) \), all components will align correctly for an accurate result.

Angle of Current to Magnetic Field

The angle between the current's direction and the magnetic field is a crucial element in determining the magnetic force exerted on the wire. In the formula \( F = I \cdot L \cdot B \cdot \sin(\theta) \), \( \theta \) represents this angle. This relationship shows that the force depends on how the wire is positioned relative to the magnetic field.

If the wire runs perpendicular to the magnetic field, as in cases (a) and (b) in our exercise, \( \theta \) is \(90^\circ\), resulting in \( \sin(90^\circ) = 1 \), maximizing the force.
In contrast, if the wire runs parallel to the field as in case (c), \( \theta \) becomes \(0^\circ\), and \( \sin(0^\circ) = 0 \), meaning no force acts on the wire.

• Perpendicular (90^\circ) maximizes force: \(\sin(90^\circ) = 1\)
• Parallel (0^\circ) negates force: \(\sin(0^\circ) = 0\)

Understanding the angle's impact on the force helps in both visualization and prediction of the magnetic force in different scenarios.