Problem 19 - Fundamentals of Physics

Question

One long wire lies along an \(x\) axis and carries a current of 30 A in the positive \(x\) direction. A second long wire is perpendicular to the \(x y\) plane, passes through the point \((0,4.0 \mathrm{~m}, 0)\), and carries a current of \(40 \mathrm{~A}\) in the positive \(z\) direction. What is the magnitude of the resulting magnetic field at the point \((0,2.0 \mathrm{~m}, 0)\) ?

Step-by-Step Solution

Answer

3.6 × 10^{-6} T.

Step 1: Identify the Components of the Magnetic Field

The problem involves two wires creating a magnetic field at a specific point. Use the right-hand rule to understand the orientation of the magnetic field created by each wire. Remember: - A current in the positive \(x\)-direction creates a magnetic field that encircles the wire in the \(yz\)-plane. - A current in the positive \(z\)-direction creates a magnetic field that encircles the wire in the \(xy\)-plane.

Step 2: Calculate the Magnetic Field from the Wire along the x-axis

The magnetic field due to a long straight wire is given by the formula: \[ B = \frac{\mu_0 I}{2 \pi r} \]where \(\mu_0 = 4\pi \times 10^{-7} \ \text{T}\cdot\text{m/A}\), \(I = 30 \ \text{A}\), and \(r = 2.0 \ \text{m}\), the separation from the point \((0, 2.0 \ \text{m}, 0)\) to the wire. Substitute the known values:\[ B_1 = \frac{4\pi \times 10^{-7} \times 30}{2 \pi \times 2} = 3 \times 10^{-6} \ \text{T} \] downward in the negative \(y\)-direction.

Step 3: Calculate the Magnetic Field from the Wire along the z-axis

For the wire carrying a current in the positive \(z\)-direction, use the same formula. Here, \(I = 40 \ \text{A}\), and the same \(r = 4.0 \ \text{m}\) from the wire to the point:\[ B_2 = \frac{4\pi \times 10^{-7} \times 40}{2 \pi \times 4} = 2 \times 10^{-6} \ \text{T} \] to the left, in the negative \(x\)-direction.

Step 4: Calculate the Magnitude of the Resultant Magnetic Field

The resultant magnetic field is found by combining the perpendicular components using the Pythagorean theorem:\[ B = \sqrt{B_x^2 + B_y^2} = \sqrt{(2 \times 10^{-6})^2 + (3 \times 10^{-6})^2} = \sqrt{4 \times 10^{-12} + 9 \times 10^{-12}} \]\[ B = \sqrt{13 \times 10^{-12}} = 3.6 \times 10^{-6} \ \text{T} \]

Step 5: Final Answer

The magnitude of the resulting magnetic field at the point \((0,2.0 \ \text{m},0)\) is \(3.6 \times 10^{-6} \ \text{T}\).

Key Concepts

Understanding the Right-Hand Rule for Magnetic Fields

The right-hand rule is a simple and effective technique used to determine the direction of a magnetic field generated by a current-carrying conductor. To apply the right-hand rule, follow these steps:

• Point your thumb in the direction of the current.
• Curl your fingers around the conductor.
• Your fingers will point in the direction of the magnetic field lines encircling the wire.

This rule is beneficial in visualizing the 3D orientation of the magnetic field. For instance, in the exercise, the current along the x-axis produces a magnetic field in the yz-plane. If you align your right thumb with the x-axis, your fingers will show that the field loops around into the yz-plane.
Similarly, for the current along the z-axis, align your thumb with the z-direction, and your fingers will demonstrate the field looping around into the xy-plane. This visualization is crucial for understanding the separate contributions of fields at a specific point in space.

Using the Long Straight Wire Formula

The formula for the magnetic field generated by a long straight wire is an essential tool in physics. This formula is derived from Ampère's Law and is given by:\[ B = \frac{\mu_0 I}{2 \pi r} \]Where:

• \(B\) is the magnetic field.
• \(\mu_0\) is the permeability of free space, approximately \(4\pi \times 10^{-7} \ \text{T}\cdot\text{m/A}\).
• \(I\) is the current through the wire.
• \(r\) is the perpendicular distance from the wire to the point of interest.

In the given problem, this formula helps calculate the magnitude and direction of the magnetic fields produced by each wire. Remember, the direction is determined separately using the right-hand rule, while the formula itself helps in calculating the magnitude. The separation distance \(r\) is critical, as it inversely affects the strength of the magnetic field.

Applying the Pythagorean Theorem in Physics

The Pythagorean theorem is a mathematical principle that also finds application in physics, especially when dealing with vector quantities like electric and magnetic fields. When two vectors are perpendicular, the resultant magnitude can be calculated using:\[ B = \sqrt{B_x^2 + B_y^2} \]This equation finds the combined effect of two perpendicular magnetic field vectors by treating them like two sides of a right triangle. In this scenario, the vectors \(B_x\) (from the z-axis wire) and \(B_y\) (from the x-axis wire) form a right angle at the point of interest, allowing us to use this theorem.
Understanding how mathematical principles such as the Pythagorean theorem apply ensures you can solve complex physics problems, especially those involving multiple, interacting fields. Recognizing when and why to apply such mathematical methods establishes a deeper grasp of the underlying physical concepts.