Problem 40 - University Physics with Modern Physics

Question

A straight, vertical wire carries a current of 1.20 A downward in a region between the poles of a large superconductingelectromagnet, where the magnetic field has magnitude \(B=\) 0.588 \(\mathrm{T}\) and is horizontal. What are the magnitude and direction of the magnetic force on a \(1.00-\mathrm{cm}\) section of the wire that is in this uniform magnetic field, if the magnetic field direction is (a) east; (b) south; (c) \(30.0^{\circ}\) south of west?

Step-by-Step Solution

Answer

(a) 0.00706 N, north; (b) 0.00706 N, west; (c) 0.00611 N, northeast.

Step 1: Understanding the Problem

To solve the problem, we need to find the magnetic force on a current-carrying wire in a magnetic field. The problem gives us the current, the length of the wire, and the magnetic field in different orientations. We will use the formula for magnetic force on a straight conductor, which is given by: \[ F = I \cdot L \cdot B \cdot \sin(\theta) \] where \( I \) 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. Each part of the problem gives us a different angle \( \theta \) to consider.

Step 2: Calculating Force for East Direction

In this case, the magnetic field is directed to the east, and the current is vertical (downward). The angle between the direction of the current (down) and the magnetic field (east) is \( 90^\circ \). Therefore, \( \sin(90^\circ) = 1 \).\[ F = 1.20 \times 0.01 \times 0.588 \times 1 = 0.007056 \; \text{N} \]The direction of the force will be determined using the right-hand rule. Using the right-hand rule, the force will be directed into the north.

Step 3: Calculating Force for South Direction

Here, the magnetic field is to the south, and the current is downward. Again, the angle between the current and the magnetic field is \( 90^\circ \), so \( \sin(90^\circ) = 1 \).\[ F = 1.20 \times 0.01 \times 0.588 \times 1 = 0.007056 \; \text{N} \]Applying the right-hand rule, the force direction will be toward the west.

Step 4: Calculating Force for 30 degrees South of West

For this scenario, the magnetic field is at an angle \( 30.0^{\circ} \) south of west. To find the correct angle \( \theta \), consider the direction perpendicular to both the wire and field directions. The effective angle with vertical is \( 90^\circ - 30^\circ = 60^\circ \). Therefore, \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \).\[ F = 1.20 \times 0.01 \times 0.588 \times \frac{\sqrt{3}}{2} \approx 0.006114 \; \text{N} \]Using the right-hand rule, the force is directed towards northeast.

Key Concepts

right-hand rule

The right-hand rule is a simple yet powerful tool to determine the direction of the magnetic force on a current-carrying wire. Here’s how you can use it:

• Imagine holding the wire with your right hand, aligning your thumb with the direction of the electric current.
• Your fingers should curl naturally in the direction of the magnetic field lines.
• The direction your palm faces indicates the direction of the magnetic force acting on the wire, perpendicular to both the current and magnetic field.

This method helps visualize the interaction between the current and magnetic field, ensuring you can easily determine the force's direction without confusion. The right-hand rule is especially useful when considering various angles of interaction, as shown in the exercise.

magnetic field

A magnetic field is an invisible field that exerts force on charged particles, such as electrons, when they move through it. The strength of this field is measured in teslas (T). In the given exercise:

• The magnetic field has a magnitude of 0.588 T.
• The direction can vary, such as east, south, or at an angle of 30 degrees south of west.

The magnetic field's direction significantly influences the magnetic force's direction on the wire. When considering the effect of the magnetic field on the wire, you must always note its orientation relative to the electric current, as this determines the angle of interaction and the resulting force.

current-carrying wire

A current-carrying wire is a conductor through which electric current flows. It generates a magnetic field around it and interacts with external magnetic fields. Here's what's essential to know about current-carrying wires:

• The current in the wire for this problem is 1.20 A downward.
• The wire segment in question is 1.00 cm long.

Current in the wire creates its own magnetic field, interacting with an external one, causing a force as described by the formula \[ F = I \cdot L \cdot B \cdot \sin(\theta) \]where each symbol represents a physical property. Understanding this interaction is key to solving problems related to electromagnetic force and designing applications like electric motors.

angle of interaction

The angle of interaction (\( \theta \)) is crucial when determining the resulting magnetic force. It is the angle between the direction of the current in the wire and the magnetic field. The force depends on the sine of this angle (\( \sin(\theta) \)), influencing its magnitude:

• A right angle (\(90^\circ\)) results in maximum force since \(\sin(90^\circ) = 1\).
• For angles other than \(90^\circ\), the force reduces, proportional to \(\sin(\theta)\).

When the exercise describes different directions of the magnetic field, like east, south, or 30 degrees south of west, it asks us to consider different angles of interaction, affecting the magnitude of the force calculated. Understanding the angle of interaction is key to solving these problems correctly.