A very long, straight wire carries a current of 0.12 A. This wire is tangent to a single-turn, c

A very long, straight wire carries a current of 0.12 A. This...

Key Concepts

Magnetic Field due to a Circular Loop

A current-carrying circular loop generates a magnetic field that is strongest at its center. The magnetic field at the center is determined by the expression B = ?0I/(2R), where I is the current through the loop and R is its radius. This principle, derived from the Biot–Savart law, is fundamental in analyzing magnetic effects in loops and coils.

Superposition Principle in Magnetism

The superposition principle states that when multiple sources of magnetic fields are present, the resultant magnetic field at any point is the vector sum of the individual fields. This principle is crucial when evaluating the net magnetic effect at a point influenced by different current-carrying conductors.

Magnetic Field due to a Straight Conductor

A long straight conductor carrying an electric current produces a magnetic field that circles around it. The magnitude of this field at a distance r from the wire is given by B = ?0I/(2?r), where ?0 is the permeability of free space and I is the current. This concept is essential for understanding how currents create magnetic fields in space.

Transcript

00:02 Okay, so we have the situation here is we have a loop, and then we have a wire tangent to the loop, which means that the top of the loop, or the middle of the loop coincides with the top of the, the top of the loop coincides with the long wire.

00:32 Now, the reason they say it's a long wire is because we don't need to worry about the endpoints.

00:38 Where this continues on to we don't know where.

00:44 And what that means is each side affects points close to it symmetrically, whereas if one was shorter, we'd have to worry about different magnetic fields from each side, but that's not the case.

00:58 We have an extremely long wire compared to the loop.

01:02 So everything is symmetrical, which means for the wire, i'll abbreviate the wire with w and the loop with l, for the wire we have our magnetic field is equal to b, the magnetic field is equal to, we have mu not times the current divided by 2 pi.

01:37 I'm sorry, 2 pi r.

01:41 And r is the distance from the wire.

01:46 In this case, we're told that at the center of this loop, we have no magnetic field.

01:52 The magnetic field here is zero.

01:56 So now how would we get that? so let's pick a direction.

02:00 It doesn't matter which direction for our current.

02:03 We have the current in red.

02:07 We'll say it's flowing to the right in the wire.

02:11 So by the right -hand rule, magnetic field has to go around like so, and that goes underneath.

02:26 So that's how that curls around.

02:28 So that means at this point here, which is straight perpendicular from the wire, the magnetic field is going to be pointing into the page.

02:40 So the magnetic field is going into the page here, and that's from the wire, so that's bw.

02:48 Now we have the loop here.

02:52 If we had the current in the loop going the same direction as the wire here, this would create a magnetic field in the same direction going into the page, but we're told that the magnetic field is zero, which means it has to be going in the opposite direction.

03:12 So we have our current, let's keep it consistent here.

03:16 We have our current flowing around counterclockwise, which means the magnetic field from the loop, bl, is flowing into the page by the right -hand rule.

03:40 So for these to cancel, the magnetic fields have to be equal.

03:45 And we have another equation for the magnetic field from a loop.

03:52 Let's call this, well, let's go right here.

03:56 The loop, we have the magnetic field is equal to.

04:02 Once again, b, you can label these.

04:08 But because these have to cancel, we can set the magnetic fields equal to each other.

04:17 Let's double line that just so it doesn't get mixed up with our equation.

04:21 So we have magnetic fields from the loop is equal to mu not times i divided by 2r, where r is the radius of the loop.

04:40 And because this line is tangent to this loop, they coincide at the same point...