Problem 76 - Fundamentals of Physics

Question

Go Figure \(29-82\) shows, in cross section, two long parallel wires spaced by distance \(d=10.0 \mathrm{~cm} ;\) each carries \(100 \mathrm{~A}\), out of the page in wire 1. Point \(P\) is on a perpendicular bisector of the line connecting the wires. In unit-vector notation, what is the net magnetic field at \(P\) if the current in wire 2 is (a) out of the page and (b) into the page?

Step-by-Step Solution

Answer

(a) Net magnetic field is \(8 \times 10^{-4} \text{ T, left}\); (b) Net magnetic field is zero.

Step 1: Understanding the Problem

We have two parallel wires, each carrying a current of 100 A. They are separated by a distance of 10.0 cm. We need to find the net magnetic field at point P, which lies on the perpendicular bisector of the line connecting the two wires, for two different scenarios: (a) when the current in wire 2 is out of the page and (b) when it is into the page.

Step 2: Magnetic Field Due to a Long Straight Wire

The magnetic field produced at a distance \(r\) by a long straight wire carrying a current \(I\) is given by the formula \(B = \frac{\mu_0 I}{2\pi r}\), where \(\mu_0 = 4\pi \times 10^{-7} \text{ Tm/A}\) is the permeability of free space.

Step 3: Calculate Magnetic Field at P for Wire 1

The distance of point P from wire 1 is \(r = \frac{d}{2} = 5.0\text{ cm} = 0.05\text{ m}\). Thus, the magnetic field at P due to wire 1 is \(B_1 = \frac{\mu_0 \times 100}{2\pi \times 0.05} \approx 4 \times 10^{-4} \text{ T}\). With current out of the page, using the right-hand rule, the magnetic field at P is directed to the left.

Step 4: Calculate Magnetic Field at P for Wire 2 (Scenario a)

In scenario (a), the current in wire 2 is also out of the page. The magnetic field at P due to wire 2, \(B_2\), will also be to the left. Using the same calculation as for wire 1, \(B_2 = 4 \times 10^{-4} \text{ T}\).

Step 5: Determine Net Magnetic Field at P (Scenario a)

Since both magnetic fields from wire 1 and wire 2 are directed to the left, the net magnetic field at P is the sum of these magnetic fields: \(B_{net} = B_1 + B_2 = 8 \times 10^{-4} \text{ T to the left}\).

Step 6: Calculate Magnetic Field at P for Wire 2 (Scenario b)

In scenario (b), the current in wire 2 is into the page. Using the right-hand rule, the magnetic field due to wire 2 at P will be directed to the right, with magnitude \(B_2 = 4 \times 10^{-4} \text{ T}\).

Step 7: Determine Net Magnetic Field at P (Scenario b)

The magnetic fields from wire 1 and wire 2 are in opposite directions. The net magnetic field at P is \(B_{net} = B_1 - B_2 = 0\). Thus, the net magnetic field is zero at point P.

Key Concepts

Right-Hand Rule

The right-hand rule is a simple tool used to determine the direction of a magnetic field relative to the direction of current flow in a wire.
Simply take your right hand and position it such that your thumb points in the direction of the current.
When you curl your fingers, the direction in which your fingers curl represents the direction of the magnetic field lines.
For a straight wire with current flowing out of the page, your thumb points out and the magnetic field circles the wire in a counterclockwise direction when viewed from above.
Conversely, if the current is into the page, the magnetic field circles the wire in a clockwise direction.
This rule is especially helpful when visualizing the scenario involving the parallel wires and how their respective magnetic fields interact.

• Use your thumb to represent the current direction.
• Your curled fingers indicate the circular magnetic field lines around the wire.
• The orientation (clockwise or counterclockwise) depends on whether the current is into or out of the page.

Perpendicular Bisector

A perpendicular bisector is a line that divides another line segment into two equal lengths and is at a right angle to it. In this problem, point P is located on the perpendicular bisector of the line connecting the two parallel wires.
This means that point P is equidistant from both wires which is essential in simplifying calculations for the magnetic fields they produce.
The location of point P ensures that its distance from each wire is half of the separation distance (d) between the wires, simplifying the use of formulas in calculating the magnetic fields.

Understanding the placement of point P relative to the wires allows us to easily calculate the individual contributions of magnetic fields from each wire, knowing they are equal in magnitude.

• Point P is equidistant from both wires.
• The field strength from each wire is calculated based on this equal distance.
• This symmetry simplifies the determination of the net magnetic field at point P.

Parallel Wires

Parallel wires carrying current interact based on the direction and magnitude of their currents. In many physics problems, parallel wires are used to explore the effects of magnetic fields.
In our exercise, two long parallel wires carry equal current. The currents' directions significantly affect the magnetic field at any point, such as point P.
When currents are in the same direction, their magnetic fields will tend to support each other.
When the currents are in opposite directions, the fields will oppose each other.
This concept of interaction can cause either a reinforcement or cancellation of magnetic fields at nearby points:

• Two parallel wires with current in the same direction attract each other.
• Two parallel wires with current in opposite directions repel each other.
• The net magnetic field depends heavily on these interactions.

In scenario (a), both currents are out of the page, meaning their magnetic fields are in the same general direction, leading to an additive effect at point P.
In scenario (b), with one current reversed, the fields counterbalance each other, resulting in a net magnetic field of zero at point P.