Problem 28 - College Physics

Question

A horizontal rod 0.200 \(\mathrm{m}\) long carries a current through a uniform horizontal magnetic field of magnitude 0.067 T that points perpendicular to the rod. If the magnetic force on this rod is measured to be \(0.13 \mathrm{N},\) what is the current flowing through the rod?

Step-by-Step Solution

Answer

The current flowing through the rod is 9.70 A.

Step 1: Identify the Known Values

First, extract and list the given values from the problem. You have the length of the rod, \( L = 0.200 \, \text{m} \), the magnetic field, \( B = 0.067 \, \text{T} \), and the force on the rod, \( F = 0.13 \, \text{N} \).

Step 2: Recall the Magnetic Force Formula

The formula to calculate the force on a current-carrying conductor in a magnetic field is given by: \[ F = IBL \sin(\theta) \] where \( F \) is the magnetic force, \( I \) is the current, \( B \) is the magnetic field strength, \( L \) is the length of the conductor, and \( \theta \) is the angle between the magnetic field and the current direction.

Step 3: Determine the Angle

Since the magnetic field is perpendicular to the rod, the angle \( \theta \) is \( 90^\circ \), and \( \sin(90^\circ) = 1 \). Substitute this into the formula to simplify it to: \[ F = IBL \]

Step 4: Solve for the Current

Rearrange the simplified formula \( F = IBL \) to solve for the current \( I \): \[ I = \frac{F}{BL} \]

Step 5: Substitute the Values

Now, substitute the known values into the formula: \[ I = \frac{0.13}{0.067 \times 0.200} \]

Step 6: Calculate the Current

Calculate the expression from the previous step: \[ I = \frac{0.13}{0.0134} = 9.70 \, \text{A} \] The current flowing through the rod is \( 9.70 \, \text{A} \).

Key Concepts

Current

The concept of current is quite vital in understanding magnetic force interactions. In simple terms, electric current is the flow of electric charge, typically described rather like water flowing through a pipe. In a metal conductor, current is formed by the movement of electrons. This process allows for the transfer of energy across the wire.

• Current is measured in amperes (A).
• It has a direction, traditionally from positive to negative.
• The current flowing through a specific area determines the magnetic force exerted on a conductor.

Understanding these basics helps to appreciate how the electrons in the rod of the exercise create such a force when interacting with the magnetic field.

Magnetic Field

A magnetic field is an invisible area around a magnetic material or an electric current that behaves as a magnet. In the context of this exercise, the magnetic field is uniform, meaning its strength and direction are constant across the area.
Magnetic fields can be thought of as lines emanating from the north to the south pole of a magnet.

• A magnetic field is measured in teslas (T).
• It has both strength and direction.
• Magnetic fields can exert forces on moving charges, like in the current-carrying rod.

The direction of the magnetic field and the current determine the force acting on the conductor, as observed in the rod scenario.

Current-Carrying Conductor

A current-carrying conductor is essentially any object that allows electric current to pass through it. In the exercise, the rod acts as a conductor.
When a conductor with current is placed in a magnetic field, it experiences a force called the magnetic force. This phenomenon is a good demonstration of the interaction between electricity and magnetism.

• The force is proportional to the length of the conductor, the current it carries, and the strength of the magnetic field.
• If the current flows perpendicular to the magnetic field, as in the exercise situation, the force is maximized.
• This force can be calculated using the formula: \( F = IBL \).

By applying this formula, students can determine the current flowing through a conductor when the force, length, and magnetic field are known.