Problem 41 - University Physics with Modern Physics

Question

A closed curve encircles several conductors. The line integral \(\oint\overrightarrow{B}\) \(\cdot\) \(\overrightarrow{dl}\) around this curve is 3.83 \(\times\) 10\(^{-4}\) T \(\cdot\) m. (a) What is the net current in the conductors? (b) If you were to integrate around the curve in the opposite direction, what would be the value of the line integral? Explain.

Step-by-Step Solution

Answer

(a) The net current is approximately 304.81 A. (b) Integrating in the opposite direction yields -3.83 × 10^{-4} T·m.

Step 1: Understanding the Context

We are given a line integral of the magnetic field \( \overrightarrow{B} \) around a closed loop, which equals \( 3.83 \times 10^{-4} \) T \( \cdot \) m. According to Ampère's Law, this integral is related to the net current \( I_{enc} \) encircled by the loop.

Step 2: Apply Ampère's Law

Ampère's Law states that \( \oint \overrightarrow{B} \cdot \overrightarrow{dl} = \mu_0 I_{enc} \), where \( \mu_0 \) is the permeability of free space (\( \mu_0 = 4\pi \times 10^{-7} \) T \cdot\ m/A). Using this formula, we can find the net current \( I_{enc} \) by rearranging to \( I_{enc} = \frac{\oint \overrightarrow{B} \cdot \overrightarrow{dl}}{\mu_0} \).

Step 3: Calculate the Net Current

Substitute the given values into the rearranged Ampère's Law: \( I_{enc} = \frac{3.83 \times 10^{-4}}{4\pi \times 10^{-7}} \). Calculate this to find the net current: \( I_{enc} = \frac{3.83 \times 10^{-4}}{1.25664 \times 10^{-6}} \approx 304.81 \) A.

Step 4: Consider Direction Change

If the curve is integrated in the opposite direction, the value of the line integral \( \oint \overrightarrow{B} \cdot \overrightarrow{dl} \) would become negative. The magnitude remains the same, so the result is \( -3.83 \times 10^{-4} \) T \cdot m.

Key Concepts

Understanding Magnetic Fields

A magnetic field is an invisible force field produced by moving electric charges, such as current flowing through a wire or motion within a magnet. It is represented by the vector field denoted as \( \overrightarrow{B} \).
Magnetic fields are fundamental in the study of electromagnetism and have both direction and magnitude. They influence magnetic materials and can induce electric currents.

• Magnetic fields exert forces on other charged particles, causing them to move in circular or spiral paths.
• They are measured in units of tesla (T).

Ampère's Law relates the magnetic field around a conductor to the electric current flowing through it, demonstrating how these forces are connected.
This understanding is crucial for solving problems related to electromagnetics, such as calculating the net current in enclosed loops.

Calculating Net Current

To determine the net current encircled by a loop, we apply Ampère's Law, which states \( \oint \overrightarrow{B} \cdot \overrightarrow{dl} = \mu_0 I_{enc} \).
Here, \( \oint \overrightarrow{B} \cdot \overrightarrow{dl} \) represents the line integral of the magnetic field around the loop, while \( \mu_0 \) is the permeability of free space, with a value of \( 4\pi \times 10^{-7} \) T·m/A.
Here's how you calculate net current:

• Rearrange the formula to solve for \( I_{enc} \), the net current, as \( I_{enc} = \frac{\oint \overrightarrow{B} \cdot \overrightarrow{dl}}{\mu_0} \).
• Substitute the given values into this equation to find the net current encircled by the loop.
• In the exercise, this calculation provides the net current of approximately 304.81 A.

By understanding how to use Ampère’s Law, you can effectively determine the current that influences the magnetic field around a conductor.

The Significance of Line Integrals

A line integral is a mathematical tool used to compute the integral of a vector field along a curve. In electromagnetism, it is often used to measure how a magnetic field interacts with a closed path.
Specifically, the line integral \( \oint \overrightarrow{B} \cdot \overrightarrow{dl} \) sums up the magnetic field over the length of the curve, considering both direction and magnitude.

• This allows us to evaluate the cumulative effect of magnetic fields around a closed loop.
• The result of a line integral depends on the direction of traversal along the curve.

In the exercise, reversing the direction of the curve changes the sign of the line integral from positive to negative, reflecting the nature of vector fields.
This is crucial for ensuring accurate measurements and predictions in work involving magnetic forces and currents.