Problem 73 - University Physics with Modern Physics

Question

Long, straight conductors with square cross sections and each carrying current \(I\) are laid side by side to form an infinite current sheet (Fig. P28.73). The conductors lie in the \(xy\)-plane, are parallel to the \(y\)-axis, and carry current in the +\(y\)-direction. There are \(n\) conductors per unit length measured along the \(x\)-axis. (a) What are the magnitude and direction of the magnetic field a distance \(a\) below the current sheet? (b) What are the magnitude and direction of the magnetic field a distance a above the current sheet?

Step-by-Step Solution

Answer

Magnitude: \(\frac{\mu_0 nI}{2}\); Direction: -z below, +z above.

Step 1: Understand the Problem

We need to find the magnetic field at a distance 'a' below and above an infinite current sheet composed of long, straight conductors. The conductors are parallel to the y-axis and carry current in the +y-direction with a density of 'n' conductors per unit length.

Step 2: Recall Biot-Savart Law

The Biot-Savart Law helps us find the magnetic field created by a current-carrying conductor. For a small segment, it is given as: \[d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I d\mathbf{l} \times \mathbf{\hat{r}}}{r^2}\]where \(d\mathbf{l}\) is the segment of the conductor carrying current \(I\), \( \mathbf{\hat{r}} \) is the unit vector pointing from the segment to the point of interest, and \(r\) is the distance from the segment to the point.

Step 3: Use Ampère's Law

To find the magnetic field due to an infinite sheet of current, use Ampère's Law. For a current sheet with a current per unit length \(K = nI\), the magnetic field \(B\) is given by: \[B = \frac{\mu_0 K}{2}\]This formula provides the magnitude of the field parallel to the sheet but perpendicular to the current flow.

Step 4: Calculate Magnetic Field Below the Sheet

Using Ampère's Law with \(K = nI\), find the magnetic field below the sheet: - The magnitude is \(B = \frac{\mu_0 nI}{2}\) for a distance 'a' below the sheet.- The direction of the magnetic field, due to the Right-Hand Rule, is in the -z direction because the current is in the +y direction.

Step 5: Calculate Magnetic Field Above the Sheet

Similarly, for the magnetic field a distance 'a' above the current sheet:- The magnitude is the same, \(B = \frac{\mu_0 nI}{2}\).- The direction of the magnetic field is in the +z direction, also determined by the Right-Hand Rule.

Step 6: Outcome

The magnitude of the magnetic field is \(\frac{\mu_0 nI}{2}\) both above and below the current sheet, but the direction is -z below the sheet and +z above the sheet.

Key Concepts

Biot-Savart Law

The Biot-Savart Law is a fundamental principle used to calculate the magnetic field generated by a current-carrying conductor. This law is particularly important when dealing with complex or small segments of current. It relates the magnetic field \( B \) at a point in space to the current \( I \), element length \( d\mathbf{l} \), and the position vector \( \mathbf{\hat{r}} \) from the element to the field point.

The expression for the magnetic field using Biot-Savart Law is given by:

• \[ d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I d\mathbf{l} \times \mathbf{\hat{r}}}{r^2} \]

This equation tells us a few important things:

• The magnetic field value \( d\mathbf{B} \) is directly proportional to the current element \( I d\mathbf{l} \).
• It’s inversely proportional to the square of the distance \( r \) between the current element and the point where the field is measured.
• The direction of \( d\mathbf{B} \) is given by the cross product between \( d\mathbf{l} \) and \( \mathbf{\hat{r}} \), resulting in a perpendicular direction to both.

The Biot-Savart Law is particularly useful in calculating the fields in systems where current paths are not symmetrically simple, necessitating precise vector calculations.

Ampère's Law

Ampère's Law provides a powerful way to calculate the magnetic field in situations with high symmetry, such as around loops and current sheets. It relates the integrated magnetic field around a closed loop to the electric current passing through the loop.

The law is mathematically expressed as:

• \[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc} \]

Where:

• \( \oint \mathbf{B} \cdot d\mathbf{l} \) is the path integral of the magnetic field around the loop.
• \( \mu_0 \) is the permeability of free space.
• \( I_{enc} \) is the current enclosed by the loop.

In practice, Ampère's Law allows us to calculate the magnetic field for arrangements such as loops, wires, or sheets where the current follows simple geometry, often allowing complex integrals to be much simplified.

For an infinite current sheet, as in the exercise, the current density per unit length \( K = nI \) is key. The law helps determine that the magnetic field magnitude on either side of the sheet is \( \frac{\mu_0 nI}{2} \), with direction dictated by the right-hand rule. This simplifies solving problems involving infinite sheets of current-carrying conductors.

current-carrying conductors

Current-carrying conductors, like wires and sheets, create magnetic fields as a result of moving charges. For any conductor through which current \( I \) flows, a magnetic field emerges around the conductor, with both magnitude and direction determined by the configurations and interactions of these moving charges.

Key characteristics of magnetic fields around conductors include:

• Direction: Determined by the Right-Hand Rule, where thumb points along the current, and the curled fingers show the magnetic field direction.
• Influence: The strength and shape of the field are influenced by the conductor's shape and the current's magnitude.
• In an infinite current sheet: The field strength is consistent and uniform on either side, as shown in problems like the textbook exercise.

Using concepts like Biot-Savart and Ampère's Law, we can predict how these factors interact to produce fields that result in forces on other nearby currents and magnetic materials. Understanding these interactions is crucial in designing electric circuits and electromagnetic devices.