Two long, parallel wires are separated by 2.0 m . Each wire has a 30 -A current, but the current

Two long, parallel wires are separated by 2.0 m . Each wire...

Two long, parallel wires are separated by 2.0 $\mathrm{m}$ . Each wire has a 30 -A current, but the currents are in opposite directions. (a) Determine the magnitude of the net magnetic field midway between the wires. (b) Determine the net magnetic field at a point 1.0 $\mathrm{m}$ to the side of one wire and 3.0 $\mathrm{m}$ from the other wire.

Two long, parallel wires are separated by 2.0 $\mathrm{m}$ . Each wire
has a 30 -A current, but the currents are in opposite directions.
(a) Determine the magnitude of the net magnetic field midway
between the wires. (b) Determine the net magnetic field at a
point 1.0 $\mathrm{m}$ to the side of one wire and 3.0 $\mathrm{m}$ from the other wire.

Key Concepts

Superposition Principle for Magnetic Fields

Magnetic fields are vector quantities, so when multiple sources contribute at a point, their fields must be added vectorially. The superposition principle allows for the calculation of the net magnetic field by summing the individual fields from each current-carrying wire, considering both their magnitudes and directions.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field around a current-carrying conductor. By pointing the thumb of the right hand in the direction of the current, the curl of the fingers shows the direction of the magnetic field lines. This rule is essential when analyzing the contributions from multiple currents, as it helps in correctly assigning the direction to each magnetic field vector.

Long Straight Wire Magnetic Field

This concept refers to the formula that gives the magnetic field produced by a long, straight conductor carrying an electric current, typically expressed as B = ??I/(2?r), where ?? is the permeability of free space, I is the current, and r is the distance from the wire. It is derived using Ampère's law and represents the fundamental behavior of magnetic fields produced by linear currents.

Transcript

00:01 This question asks about two wires parallel to each other, and they're separated by two meters.

00:15 So i'm going to say that i'm just going to write two meters here.

00:21 It's going to ask us about the halfway point.

00:24 So i'm going to set r is half of two meters, so that would be one meter.

00:31 The current in each one is 30 amps, but the current is flowing in opposite directions.

00:43 And that's important because it asks us about the magnetic field halfway between.

00:50 Well, if they were flowing in the same direction, then according to the right -hand rule, they would cancel each other out, and the net magnetic field would be zero halfway between.

01:00 But since they're flowing in opposite directions, then we can just say that the magnetic field is two times the magnetic field of one of the wires.

01:11 So b equals mu zero over two pi times i over r.

01:24 But we need to take this times two because there are two wires.

01:29 So we can just put that in a calculator.

01:36 I like to use desmos .com on the computer.

01:42 Mew sub zero is 4 pi.

01:46 Let me write that down for you.

01:49 So this is for air times 10 to the negative 7th power, tesla meters per amp.

02:03 4 times pi.

02:09 Times 10 to the negative seventh power.

02:13 So that's mu zero.

02:15 And i'm just gonna divide that by two pi.

02:23 And then i'm going to multiply that by i over r.