Problem 35 - University Physics with Modern Physics

Question

A closed curve encircles several conductors. The line integral \(\phi \overrightarrow{\boldsymbol{B}} \cdot d \vec{l}\) around this curve is \(3.83 \times 10^{-4} \mathrm{T} \cdot \mathrm{m} .\) (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) 303.4 A; (b) Line integral is \(-3.83 \times 10^{-4} \mathrm{T} \cdot \mathrm{m}\).

Step 1: Use Ampère's Law

According to Ampère's Law, the line integral of the magnetic field \( \overrightarrow{\boldsymbol{B}} \) around a closed path is equal to \( \mu_0 \) times the net current \( I \) enclosed by the path. The formula for this is: \( \oint \overrightarrow{\boldsymbol{B}} \cdot d \vec{l} = \mu_0 I \).Given that \( \oint \overrightarrow{\boldsymbol{B}} \cdot d \vec{l} = 3.83 \times 10^{-4} \mathrm{T} \cdot \mathrm{m} \), we can use this equation to find the net current, \( I \).

Step 2: Solve for Net Current

We know that:\[ \oint \overrightarrow{\boldsymbol{B}} \cdot d \vec{l} = \mu_0 I \]Solving for \( I \), we have:\[ I = \frac{\oint \overrightarrow{\boldsymbol{B}} \cdot d \vec{l}}{\mu_0} \]Assuming \( \mu_0 = 4\pi \times 10^{-7} \text{ T} \cdot \text{m/A} \), substitute \( 3.83 \times 10^{-4} \text{ T} \cdot \text{m} \) for the line integral:\[ I = \frac{3.83 \times 10^{-4}}{4\pi \times 10^{-7}} \approx 303.4 \text{ A} \].

Step 3: Consider the Opposite Direction Integral

If the integration is performed in the opposite direction, this simply inverts the sign of the line integral due to the reversal of path direction. Thus, the value becomes \(-3.83 \times 10^{-4} \mathrm{T} \cdot \mathrm{m} \). Ampère's Law itself remains unchanged as it is a scalar relation dependent on the enclosed current and measures direction change with a negative sign.

Key Concepts

Magnetic Field

A magnetic field is a region around a magnetic material or a moving electric charge within which the force of magnetism acts. It is represented by the symbol \( \overrightarrow{\boldsymbol{B}} \), and it's a vector field.

• Think of the magnetic field lines like invisible threads pulling magnetic objects towards or away from a magnet.
• These lines give us a visual cue about the strength and direction of the magnetic influence.

In the context of this problem, we are particularly interested in how the magnetic field interacts with electrical currents. When a current flows through conductors, it generates a magnetic field around them. This relationship is fundamental in electromagnetism and ties directly into Ampère's Law. Understanding the interaction between currents and magnetic fields is essential in fields like electrical engineering and physics.The strength of a magnetic field can be expressed in teslas \( \text{T} \), which measure the force on moving charges within the field. It's critical to understand how these fields contribute to practical applications like electric motors, MRI machines, and more.

Line Integral

In mathematics, a line integral is a type of integral where a function is integrated along a curve. In physics, particularly electromagnetism, it describes how a field behaves over a path or boundary. For the magnetic field \( \overrightarrow{\boldsymbol{B}} \), we use a line integral to understand its cumulative effect across a closed loop.

• The line integral is computed using \( \oint \overrightarrow{\boldsymbol{B}} \cdot d \vec{l} \), representing the integral of the magnetic field over the path \( d \vec{l} \).
• This integral helps in calculating the total magnetic effect along a given path around current-carrying conductors.

When we perform the line integral of a magnetic field around a closed curve, according to Ampère's Law, it gives us the net current encircled by that path. If the direction of integration changes, the integral’s sign is reversed, implying that the line integral is direction-sensitive. Thus, flipping the integration path reverses the final sign of the calculated line integral, but the fundamental relationship described by Ampère's Law remains unaffected.

Net Current

Net current indicates the total current flowing through conductors encircled by a chosen closed path. Ampère's Law provides the concept that relates the magnetic field around a path to the current enclosed by it.

• According to Ampère's Law, \( \oint \overrightarrow{\boldsymbol{B}} \cdot d \vec{l} = \mu_0 I \), where \( \mu_0 \) is the permeability of free space, a constant. By solving this equation for \( I \), we find the net current.
• In our exercise, the given line integral allowed us to compute the enclosed current as approximately 303.4 A.

Permeability of free space \( \mu_0 \) is critical to understanding the relationship between magnetic fields and currents. Its value helps us convert the magnetic effects observed around conductors into quantifiable current values. This principle is crucial in designing and understanding electrical circuits, power systems, and many related technologies. Understanding the net current’s calculation helps engineers and scientists design more efficient magnetic and electromagnetic systems.