Magnetic Force on Current-Carrying Wire: Videos & Practice Problems

When a current-carrying wire is placed in a magnetic field, it experiences a force. This force can be calculated using the equation F = BILsin(θ), where F is the force, B is the magnetic field strength, I is the current, L is the length of the wire, and θ is the angle between the current and the magnetic field. The right-hand rule helps determine the direction of the force, which can cause the wire to bend. Understanding these principles is crucial in electromagnetism.

Magnetic Force on Current-Carrying Wire

When a current-carrying wire is placed within a magnetic field, it experiences a force due to the interaction between the magnetic field and the moving charges within the wire. This phenomenon is rooted in the principles of electromagnetism, where moving charges generate a magnetic field and, when subjected to an external magnetic field, feel a force. The fundamental equation governing this interaction is given by:

$$ F = BIL \sin(\theta) $$

In this equation, F represents the magnetic force acting on the wire, B is the magnetic field strength, I is the current flowing through the wire, L is the length of the wire within the magnetic field, and θ is the angle between the direction of the current and the magnetic field. The right-hand rule is a useful tool for determining the direction of the force: point your thumb in the direction of the current, your fingers in the direction of the magnetic field, and your palm will face the direction of the force.

For example, if a wire carrying a current is placed in a magnetic field directed into the page, the force on the wire can be determined by applying the right-hand rule. If the current flows upwards, the force will be directed into the page, while reversing the current direction will reverse the force direction, pushing it out of the page.

To find the strength of the magnetic field when given the force, current, and length of the wire, you can rearrange the equation to:

$$ B = \frac{F}{IL \sin(\theta)} $$

In many cases, if the current and magnetic field are perpendicular, the sine of 90 degrees is 1, simplifying the calculation. For instance, if a 2-meter long wire experiences a force of 3 newtons with a current of 4 amperes, the magnetic field strength can be calculated as:

$$ B = \frac{3 \, \text{N}}{4 \, \text{A} \times 2 \, \text{m}} = 0.375 \, \text{T} $$

Understanding the direction of the force is crucial, especially when analyzing scenarios where the force acts downward. By applying the right-hand rule, if the force is directed downwards and the magnetic field is into the page, the current must flow to the left.

This interplay between current, magnetic fields, and forces is foundational in electromagnetism and has practical applications in electric motors, generators, and various electromagnetic devices.

Find Force on Current-Carrying Wire at an Angle

In this scenario, we analyze the force exerted on a wire carrying an electric current within a magnetic field. The wire has a length of 2 meters, a magnetic field strength of 3 teslas, and carries a current of 4 amps. The magnetic field is directed along the negative y-axis.

The force on the wire can be calculated using the formula:

$$ F = B \cdot I \cdot L \cdot \sin(\theta) $$

where:

• F is the force in newtons (N),
• B is the magnetic field strength in teslas (T),
• I is the current in amperes (A),
• L is the length of the wire in meters (m),
• θ is the angle between the magnetic field and the current direction.

In the first case, both the magnetic field and the current are parallel (both directed downwards), resulting in an angle of 0 degrees. Since the sine of 0 degrees is 0, the force is:

$$ F = B \cdot I \cdot L \cdot \sin(0) = 0 $$

This indicates that there is no force acting on the wire when the current and magnetic field are parallel.

In the second scenario, the magnetic field remains directed downwards, while the current flows horizontally along the positive x-axis. Here, the angle between the magnetic field and the current is 90 degrees. The sine of 90 degrees is 1, so the force can be calculated as:

$$ F = B \cdot I \cdot L \cdot \sin(90) = 3 \cdot 4 \cdot 2 \cdot 1 = 24 \text{ N} $$

To determine the direction of the force, we apply the right-hand rule: with the magnetic field pointing down and the current to the right, the force is directed into the page.

In the third case, the magnetic field is still directed downwards, but the current makes an angle of 53 degrees with the y-axis. The angle between the magnetic field and the current can be calculated as follows:

• Counterclockwise from the magnetic field to the current gives an angle of 127 degrees.
• Alternatively, going clockwise gives an angle of 233 degrees.

Regardless of the approach, we can use the sine of 127 degrees to find the force:

$$ F = B \cdot I \cdot L \cdot \sin(127) = 3 \cdot 4 \cdot 2 \cdot \sin(127) $$

Calculating this yields a force of approximately 19.2 N. The direction of the force depends on the current's direction:

• If the current flows in one direction, the force is directed out of the page.
• If the current flows in the opposite direction, the force is directed into the page.

This illustrates how the orientation of the current relative to the magnetic field affects the direction of the force, while the magnitude remains consistent regardless of the current's direction.

A 5-m current-carrying wire (red line) is ran through a 4 T magnetic field (blue lines), as shown. The angle shown is 30°. What must the magnitude and direction of the current in the wire be when it feels a 3 N force directed into the page?

• A. 0.17 A @ 30° below the +x-axis
• B. 0.17 A @ 30° above the −x-axis
• C. 0.30 A @ 30° below the +x-axis
• D. 0.30 A @ 30° above the −x-axis
• E. 0.49 A @ 30° below the +x-axis
• F. 0.49 A @ 30° above the −x-axis
• G. 3.25 A @ 30° below the +x-axis
• H. 3.25 A @ 30° above the −x-axis

Answer: (A)

Here’s what students ask on this topic:

What is the formula for the magnetic force on a current-carrying wire?

The formula for the magnetic force on a current-carrying wire is given by:

$F=BIL\sin \left(\theta \right)$

where:

• $F$ is the magnetic force
• $B$ is the magnetic field strength
• $I$ is the current
• $L$ is the length of the wire
• $\theta$ is the angle between the current and the magnetic field

How do you use the right-hand rule to determine the direction of the magnetic force on a current-carrying wire?

To use the right-hand rule to determine the direction of the magnetic force on a current-carrying wire, follow these steps:

1. Point your thumb in the direction of the current (I).
2. Point your fingers in the direction of the magnetic field (B).
3. Your palm will face the direction of the magnetic force (F).

For example, if the current is flowing upwards and the magnetic field is directed to the right, your palm will face out of the page, indicating the force is directed outwards.

What factors affect the magnitude of the magnetic force on a current-carrying wire?

The magnitude of the magnetic force on a current-carrying wire is affected by the following factors:

• The strength of the magnetic field (B): A stronger magnetic field increases the force.
• The current (I) in the wire: A higher current increases the force.
• The length of the wire (L) within the magnetic field: A longer wire increases the force.
• The angle (θ) between the current and the magnetic field: The force is maximized when the angle is 90 degrees (sin(90°) = 1).

Why does a current-carrying wire experience a force in a magnetic field?

A current-carrying wire experiences a force in a magnetic field due to the interaction between the magnetic field and the moving charges within the wire. When charges move through a magnetic field, they experience a force given by $F=qvB\sin \left(\theta \right)$. In a wire, these moving charges constitute a current, and the collective force on these charges results in a force on the wire itself, described by $F=BIL\sin \left(\theta \right)$.

How can you calculate the magnetic field strength if the force, current, and length of the wire are known?

To calculate the magnetic field strength (B) if the force (F), current (I), and length of the wire (L) are known, you can rearrange the formula for the magnetic force:

$F=BIL\sin \left(\theta \right)$

Solving for B gives:

$B=\frac{F}{/}$

Plug in the values for F, I, L, and θ to find the magnetic field strength.