Problem 30 - College Physics

Question

\(\bullet\) Between the poles of a powerful magnet is a cylindrical uniform magnetic field with a diameter of 3.50 \(\mathrm{cm}\) and a strength of 1.40 \(\mathrm{T}\) . A wire carries a current through the center of the field at an angle of \(65.0^{\circ}\) to the magnetic field lines. If the wire experiences a magnetic force of \(0.0514 \mathrm{N},\) what is the current flowing in it?

Step-by-Step Solution

Answer

The current flowing in the wire is approximately 1.14 A.

Step 1: Understand the Formula

The magnetic force on a wire in a uniform magnetic field is given by the formula: \( F = B \cdot I \cdot L \cdot \sin(\theta) \), where \( F \) is the magnetic force, \( B \) is the magnetic field strength, \( I \) is the current, \( L \) is the length of the wire within the magnetic field, and \( \theta \) is the angle between the wire and the magnetic field.

Step 2: Solve for Current (\( I \))

Rearrange the formula to solve for the current \( I \): \[ I = \frac{F}{B \cdot L \cdot \sin(\theta)}.\]

Step 3: Determine the Length of Wire (\(L\))

Since the cross-section of the magnetic field is circular with a diameter of 3.50 cm, the length \( L \) of the wire within the magnetic field is equal to the diameter of the circle, which is 0.035 m.

Step 4: Calculate Sin of the Angle (\(\sin(65.0^\circ)\))

Calculate \( \sin(65.0^\circ) \): \[\sin(65.0^\circ) \approx 0.9063.\]

Step 5: Substitute Values and Calculate Current

Substitute the known values into the equation: \[ I = \frac{0.0514}{1.40 \times 0.035 \times 0.9063}.\]Calculate to find \( I \).

Step 6: Perform the Calculation

Carry out the calculation:\[ I = \frac{0.0514}{1.40 \times 0.035 \times 0.9063} \approx 1.14 \,\mathrm{A}.\]

Key Concepts

Magnetic Field

The concept of a magnetic field can be quite fascinating. It's like an invisible force field that surrounds magnets and electric currents. When we talk about the magnetic field in physics, we're referring to the region around a magnet where its magnetic force can influence moving charges or other magnets. This field is represented by the symbol \( B \) and is measured in teslas (T).
The strength of a magnetic field can vary depending on the source of the field. In this particular exercise, we have a uniform magnetic field, meaning the field strength is the same at every point within the field.

• Uniform Magnetic Field: The consistent strength of the magnetic field throughout its region.
• Magnetic Field Lines: Imaginary lines that represent the direction and strength of a magnetic field.

If you imagine a magnet, the field lines emerge from the north pole and re-enter at the south pole. These lines help us understand and visualize the way the magnetic force operates around the wire in the problem. They also indicate the direction in which a force would act on a moving charge within the field.

Current Calculation

Current calculation in physics is how we determine the flow of electric charge through a conductor. In this particular problem, we are calculating the current flowing through a wire when subjected to a magnetic force. The formula given for magnetic force, \( F = B \cdot I \cdot L \cdot \sin(\theta) \), is pivotal for solving these kinds of problems. Here's a breakdown:

• \( F \): Represents the force exerted on the wire by the magnetic field.
• \( B \): Magnetic field strength, a measure of how powerful the magnetic field is.
• \( I \): The current, or the rate of flow of charge through the wire.
• \( L \): Length of wire exposed to the magnetic field.
• \( \sin(\theta) \): Trigonometric function that accounts for the angle between the wire and the magnetic field.

To find the current \( I \), we rearrange the formula to: \[ I = \frac{F}{B \cdot L \cdot \sin(\theta)} \]This equation allows us to determine how much charge is being pushed through the wire due to the influence of the magnetic field, given the known magnetic force and the geometry of the situation.

Trigonometry in Physics

In physics, trigonometry allows us to calculate forces that are not aligned along the same direction. The concept of angles is important in scenarios where forces or fields are not purely horizontal or vertical. In our exercise, trigonometry helps in determining the effective component of the magnetic force acting along the wire.
Understanding trigonometric functions like sine (\( \sin \)) is crucial as they let us capture the influence of angles in vector-sensitive operations, such as the magnetic force on a wire.

• Sine Function: \( \sin(\theta) \) is a commonly used trigonometric function in physics, particularly when dealing with forces at angles.
• Angle of Interaction: In the problem, the wire forms a \( 65^\circ \) angle with the magnetic field, affecting how much of the magnetic force influences the current.

Calculating \( \sin(65.0^\circ) \) which is approximately 0.9063, helps us adjust the formula to account for how the force is distributed along the wire. This is inherently the trigonometric aspect of factoring in angles during calculations that influence physical outcomes.