Two infinitely long, straight wires are parallel and separated by a distance of one meter. They

Two infinitely long, straight wires are parallel and...

Key Concepts

Equilibrium Point (Field Cancellation)

In setups where multiple magnetic fields overlap, an equilibrium point may exist where the net magnetic field is zero. Determining this point involves solving for distances where the contributions from each wire, adjusted for both magnitude (via the inverse relationship with distance) and direction (via the right-hand rule), exactly cancel out. This concept underlies many practical applications in electromagnetic field analysis.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field produced by a current-carrying wire. By pointing the thumb in the direction of the current, the curled fingers indicate the circulation of the magnetic field, which is vital to understanding the orientation and subsequent vector addition of fields from multiple wires.

Magnetic Field of an Infinite Wire

A long, straight current-carrying wire produces a magnetic field whose magnitude is given by the formula B = ?0*I/(2?r), where I is the current and r is the perpendicular distance from the wire. This relationship is fundamental in determining how the magnetic field decreases with distance and is derived from Ampère’s law, highlighting that the field encircles the wire in concentric circles.

Superposition Principle

The superposition principle states that when multiple sources produce magnetic fields, the net magnetic field at any given point is the vector sum of the individual fields. This concept is essential in problems where several wires with different currents generate overlapping fields, as it allows for the cancellation or reinforcement of field strengths, depending on their magnitudes and directions.

Transcript

00:01 In this problem, we're given that there's two wires, two infinitely long wires, carrying current in the same direction, separated by a particular distance.

00:11 Now, let's just give a little drawing of this.

00:13 So we'd have, let's draw them from the cross -sectional side.

00:17 So say we have one wire here, a wire one, and it's going to have a current of four times, what we'll call i, because that is the current that is going to be going through wire two.

00:31 Now, the current being carried in the same direction.

00:33 So i'll just say the current's going into the page, and we'll call that an x heading away from us.

00:40 We know and are given that the distance between the two wires is one meter.

00:47 And we're asked along a line perpendicular to both of these, so connecting all of them.

00:53 Then we'll draw a little line here, a little dotted line, connecting them.

00:58 The question is, where along this line, anywhere is the magnetic field, net magnetic field from both wires going to be zero.

01:09 Now, that is an important question.

01:13 Well, actually, in order to answer that question, an important thing is to consider where the magnetic fields are allowed to cancel and where they're not going to cancel in the regions around these wires.

01:26 So really we have to consider several spaces and think about how the magnetic fields are going to add up, those look places.

01:36 So we're going to have a region that lies between the two wires.

01:39 We have to consider what the magnetic fields are going to do there.

01:42 And then we have a region to the right of wire two we have to consider and then the region to the left of wire one that we have to consider.

01:49 Now before we can decide whether or not the magnetic fields are going to cancel or not, we have to draw what the direction of the magnetic fields are going to be.

01:56 If we follow the right hand rule and put our thumb into the screen along the direction of the currents, then our fingers are going to wrap around in a clockwise manner.

02:06 So both of these wires are going to be creating magnetic fields, circular magnetic fields around them, that are clockwise in nature.

02:22 So this would be, say, the magnetic field from wire one, but that's going to vary with distance.

02:29 And this is going to be the magnetic field from wire two, and that's going to vary with distance.

02:34 Now, if we look at the regions here, so in the region to the left of wire two, the magnetic field is always heading up, the magnetic field due to wire 1 is always heading up, and the magnetic field due to wire 2 is always heading up.

02:47 So the magnetic fields are always going to add to the left here.

02:53 So that's not a region where the magnetic fields can cancel.

02:55 And we see the same thing happens to the right of wire 2.

02:59 The magnetic fields will always add because they're always heading downward along that dotted blue line.

03:04 Now in between the two wires we see that the magnetic field due to wire 1 is heading down, and the magnetic field due to wire 2 is heading up.

03:12 So this is the region where the magnetic fields can cancel.

03:17 And so that's the region we're going to focus on.

03:19 So we need to find a location between these two where the magnetic fields will cancel...