A current Loop A B C D is held fixed on the plane of the...

A current Loop $A B C D$ is held fixed on the plane of the paper as shown in the figure. The arcs $\mathrm{BC}$ (radius $=\mathrm{b}$ ) and $\mathrm{DA}$ (radius $=\mathrm{a}$ ) of the Loop are joined by two straight wires $\mathrm{AB}$ and CD. A steady current $I$ is flowing in the Loop. angle made by $\mathrm{AB}$ and $\mathrm{CD}$ at the origin is $30^{\circ}$. Another straight thin wire with steady current $\mathrm{I}_{1}$ flowing out of the plane of the paper is kept at the origin. Due to the presence of the current $\mathrm{I}_{1}$ at the origin (a) The forces on $\mathrm{AB}$ and $\mathrm{DC}$ are zero. (b) The forces on $A D$ and $B C$ are zero. (c) The magnitude of the net force on the loop is given by $\left[\left(\mu_{0} I I_{1}\right) /(4 \pi)\right][2(b-a)+(\pi / 3)(a+b)]$ (d) The magnitude of the net force on the loop is given by $\left[\left(\mu_{0} I \bar{I}_{1}\right) /(24 \mathrm{ab})\right](\mathrm{b}-\mathrm{a})$

A current Loop $A B C D$ is held fixed on the plane of the paper as shown in the figure. The arcs $\mathrm{BC}$ (radius $=\mathrm{b}$ ) and $\mathrm{DA}$ (radius $=\mathrm{a}$ ) of the Loop are joined by two straight wires $\mathrm{AB}$ and CD. A steady current $I$ is flowing in the Loop. angle made by $\mathrm{AB}$ and $\mathrm{CD}$ at the origin is $30^{\circ}$. Another straight thin wire with steady current $\mathrm{I}_{1}$ flowing out of the plane of the paper is kept at the origin.
Due to the presence of the current $\mathrm{I}_{1}$ at the origin
(a) The forces on $\mathrm{AB}$ and $\mathrm{DC}$ are zero.
(b) The forces on $A D$ and $B C$ are zero.
(c) The magnitude of the net force on the loop is given by $\left[\left(\mu_{0} I I_{1}\right) /(4 \pi)\right][2(b-a)+(\pi / 3)(a+b)]$
(d) The magnitude of the net force on the loop is given by $\left[\left(\mu_{0} I \bar{I}_{1}\right) /(24 \mathrm{ab})\right](\mathrm{b}-\mathrm{a})$

Key Concepts

Force on a Current Loop

When a closed loop carrying current is placed in an external magnetic field, each segment of the loop experiences a Lorentz force. The net force on the loop is found by vectorially summing the contributions from all segments, taking into account both magnitude and direction of the forces, which is an important concept in electromagnetic force analysis.

Superposition Principle in Magnetostatics

This principle indicates that the total magnetic field at any point is the vector sum of the fields produced by individual current elements. It allows for the independent calculation of magnetic field contributions from various parts of a system and is vital when analyzing complex current configurations.

Lorentz Force Law

This principle states that a current-carrying conductor placed in a magnetic field experiences a force given by F = I (L × B), where I is the current, L is the length (or vector direction) of the conductor, and B is the magnetic field. It is essential for analyzing how currents interact with external magnetic fields.

Magnetic Field from a Current-Carrying Wire

A wire carrying a steady current generates a magnetic field that circles the wire, with the magnitude decreasing as the distance increases, typically described by Ampère's Law or the Biot–Savart Law. Understanding this concept is crucial when evaluating the magnetic environment created by an external current source.

Transcript

00:02 In this example, we're going to string up a conductive loop of wire here in a pendulum -like situation where it swings down into this region where there is a uniform constant magnetic field.

00:17 And because it swings into a region where there is a magnetic field, the flux through the loop is going to change.

00:25 And by faraday's law of induction, there will be an induced emf in the loop and therefore an induced current in the loop.

00:32 And what we want to figure out is what is the direction of the current flowing through the loop at point a and point b, as i've marked on the diagram here, when it's swinging from left to right, and when it's swinging from right to left.

00:49 So first we're going to look at left to right.

00:53 And we're going to talk about the direction in terms of these points, x, y, and z that we've marked on our loop.

00:59 So we'll either say that the current is flowing from x to y to z or z to y to x.

01:07 But ultimately it's just clockwise or counterclockwise.

01:12 Okay.

01:13 So when the pendulum is going left to right, it's coming down like this and entering the magnetic field.

01:21 We can see that the flux is going to be increasing.

01:27 So flux is going up.

01:30 And we know that the direction of the flux is going to be in the direction of positive x, we'll call it.

01:42 So we'll call this direction plus x and this direction minus x.

01:49 And since we know the flux is increasing in that direction, a current's going to be created that will have a magnetic field that it poses that direction.

01:58 So we want to create a magnetic field in minus x.

02:05 So we want b induced in minus x.

02:16 And to do that, if we point our thumb in the direction of the induced field we want, and curl our hand around, as our fingers slightly curl, we can see that the direction is going to be from x to y to z, because our fingers will point in the direction of the current.

02:41 So x to y to z.

02:47 So we're just going to use the right -hand rule to determine all of these now.

02:53 Okay.

02:55 So now at point b, at this point it's going to be coming out of the magnetic field.

03:02 So the flux will be going down if we're exiting the field.

03:09 So flux is going down and the direction of the field is again plus x.

03:18 So since the flux is going down, we will want to create a magnetic field that has the direction the same as the current magnetic field that's creating our flux...

Please give Ace some feedback