Problem 66 - Fundamentals of Physics

Question

Two long wires lie in an \(x y\) plane, and each carries a current in the positive direction of the \(x\) axis. Wire 1 is at \(y=10.0 \mathrm{~cm}\) and carries \(6.00 \mathrm{~A}\); wire 2 is at \(y=5.00 \mathrm{~cm}\) and carries \(10.0 \mathrm{~A}\). (a) In unitvector notation, what is the net magnetic field \(\vec{B}\) at the origin? (b) At what value of \(y\) does \(\vec{B}=0 ?(\mathrm{c})\) If the current in wire 1 is reversed, at what value of \(y\) does \(\vec{B}=0 ?\)

Step-by-Step Solution

Answer

(a) \( 5.2 \times 10^{-5} \hat{k} \text{ T} \); (b) \( y = 17.5 \text{ cm} \); (c) \( y = 8.125 \text{ cm} \).

Step 1: Magnetic Field Due to a Current-Carrying Wire

First, let's recall the formula for the magnetic field produced by a long straight wire carrying a current, which is given by \[ \vec{B} = \frac{\mu_0 I}{2\pi r} \hat{k} \]where \( \mu_0 \) is the permeability of free space, \( I \) is the current, \( r \) is the perpendicular distance from the wire, and \( \hat{k} \) is the unit vector in the direction perpendicular to the plane containing the wire and the point where the field is being measured.

Step 2: Calculate the Magnetic Field at the Origin from Wire 1

Wire 1 is positioned at \( y = 10.0 \text{ cm} = 0.10 \text{ m} \), carries a current of \( I_1 = 6.00 \text{ A} \). The origin is below wire 1, so the distance \( r_1 = 0.10 \text{ m} \). The magnetic field due to wire 1 at the origin is:\[ \vec{B}_1 = \frac{\mu_0 I_1}{2\pi r_1} \hat{k} = \frac{(4\pi \times 10^{-7} \text{ Tm/A}) \times 6.00}{2\pi \times 0.10} \hat{k} = 1.2 \times 10^{-5} \text{ T} \hat{k} \]The direction of \( \vec{B}_1 \) is out of the page (using the right-hand rule).

Step 3: Calculate the Magnetic Field at the Origin from Wire 2

Wire 2 is at \( y = 5.00 \text{ cm} = 0.05 \text{ m} \), carrying a current of \( I_2 = 10.0 \text{ A} \). The distance from the origin is \( r_2 = 0.05 \text{ m} \). The magnetic field due to wire 2 is:\[ \vec{B}_2 = \frac{\mu_0 I_2}{2\pi r_2} \hat{k} = \frac{(4\pi \times 10^{-7} \text{ Tm/A}) \times 10.0}{2\pi \times 0.05} \hat{k} = 4.0 \times 10^{-5} \text{ T} \hat{k} \]The direction of \( \vec{B}_2 \) is also out of the page.

Step 4: Calculate Net Magnetic Field at the Origin

The net magnetic field at the origin, \( \vec{B}_{net} \), is the sum of the magnetic fields from wires 1 and 2:\[ \vec{B}_{net} = \vec{B}_1 + \vec{B}_2 = 1.2 \times 10^{-5} \hat{k} + 4.0 \times 10^{-5} \hat{k} = 5.2 \times 10^{-5} \hat{k} \text{ T} \] Therefore, the net magnetic field at the origin is \( 5.2 \times 10^{-5} \text{ T} \hat{k} \).

Step 5: Determine y where Magnetic Field is Zero (Same Current Direction)

For \( \vec{B} = 0 \), the magnetic fields from the two wires must cancel each other:\[ \frac{6}{y - 0.10} = \frac{10}{y - 0.05} \]Simplifying gives: \[ 6(y - 0.05) = 10(y - 0.10) \]\[ 6y - 0.30 = 10y - 1.0 \]Solving for \( y \): \[ 4y = 0.70 \]\[ y = 0.175 \text{ m} \text{ or } 17.5 \text{ cm} \] Thus, \( \vec{B} = 0 \) at \( y = 17.5 \text{ cm} \).

Step 6: Determine y where Magnetic Field is Zero (Reversed Current in Wire 1)

If the current in wire 1 is reversed, the magnetic fields need to add up to zero:\[ \frac{6}{y - 0.10} = \frac{10}{y - 0.05} \]Now assuming wire 1's field is reversed, \[ \frac{6}{y - 0.10} + -\frac{10}{y - 0.05} = 0 \]Rearranging, we get:\[ 6(y - 0.05) = -10(y - 0.10) \]\[ 6y - 0.30 = -10y + 1.0 \]Solving gives: \[ 16y = 1.30 \]\[ y = 0.08125 \text{ m} \text{ or } 8.125 \text{ cm} \] Hence, \( \vec{B} = 0 \) at \( y = 8.125 \text{ cm} \) after reversing the current in wire 1.

Key Concepts

Ampere's Law

Ampere's Law is a fundamental principle in electromagnetism that describes how an electric current generates a magnetic field. It states that the line integral of the magnetic field \( \vec{B} \) around any closed loop is equal to the permeability of free space \( \mu_0 \) times the total current \( I \) passing through the loop. Mathematically, it is expressed as: \[ \oint \vec{B} \cdot d\vec{l} = \mu_0 I \] where \( d\vec{l} \) is a differential element of the closed loop.
Ampere's Law helps in calculating the magnetic field in situations with high symmetry, such as a long straight wire. In the context of the exercise, it helps us understand how each wire affects the magnetic field at a given point, like the origin.
### Practical Application For a straight, long wire, Ampere's Law simplifies to an expression for the magnetic field around the wire, as calculated using the right-hand rule. This calculation assumes cylindrical symmetry around the wire, making it applicable for cases like two wires running parallel, as we see in the exercise.

right-hand rule

The right-hand rule is a simple way to determine the direction of the magnetic field created by a current-carrying wire. It involves using the orientation of your right hand to predict the magnetic field's direction:

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

In the exercise, this rule is used to deduce that both wires create magnetic fields pointing in the same direction (out of the page) at the origin. By reversing the current in one of the wires, the field direction from that wire would reverse too, leading to the conditions where the fields can cancel each other out, achieving a net magnetic field of zero at certain points along the y-axis.

permeability of free space

The permeability of free space, often represented as \( \mu_0 \), is a physical constant that describes the extent to which a magnetic field can penetrate and influence the vacuum of free space. It has a value of \( \mu_0 = 4\pi \times 10^{-7} \text{ Tm/A} \). This constant is fundamental in equations involving magnetic fields and is crucial in the understanding of electromagnetic forces.
### Role in Magnetic Field Calculation In the given exercise, \( \mu_0 \) is a key part of the equation used to calculate the magnetic fields \( \vec{B}_1 \) and \( \vec{B}_2 \) produced by the wires. It quantifies how the current interacts with the space around it to produce a magnetic field. The greater the permeability, the stronger the magnetic field for a given current and distance from the wire.
Understanding \( \mu_0 \) is essential for accurately determining the magnetic influence a current exerts in any given space, a concept that is often examined with Ampere's Law and the right-hand rule.