Use Eqs. (8.67) and (8.37) to determine both the electric...

Name: Problem 12, Chapter 8: Clifford algebra: a computational tool for physicists
Uploaded: 2021-07-25T14:38:08-08:00
Duration: 7 min 12 s
Channel: Vishal Gupta
Description: Use Eqs. (8.67) and (8.37) to determine both the electric and magnetic dipoles of the free electron. Observe that the electric dipole is induced by the motion and is perpendicular to both the direction of the magnetic dipole and the direction of motion.

Use Eqs. (8.67) and (8.37) to determine both the electric and magnetic dipoles of the free electron. Observe that the electric dipole is induced by the motion and is perpendicular to both the direction of the magnetic dipole and the direction of motion.

Key Concepts

Magnetic Dipole Moment

The magnetic dipole moment is a measure of the strength and orientation of an object's magnetic property. In classical electrodynamics, it is associated with a current loop or with the intrinsic angular momentum (spin) of a charged particle, such as an electron. For an electron, this moment arises naturally from its charge in motion (either orbital or intrinsic spin), making it behave like a tiny bar magnet whose magnetic field can interact with external magnetic fields.

Electric Dipole Moment

The electric dipole moment generally describes a separation of positive and negative charges, creating a directional electric field. For a free electron, which in its rest frame has no intrinsic electric dipole, an electric dipole moment can be induced due to its motion. This induced dipole is produced by the Lorentz transformation of the electromagnetic fields; as the electron moves, its magnetic dipole moment gives rise to an additional electric field component that acts like a perpendicular electric dipole.

Relativistic Transformation of Electromagnetic Fields

The relativistic transformation of electromagnetic fields explains how electric and magnetic fields mix when viewed from different inertial frames. Through the Lorentz transformations, a magnetic dipole in one frame can induce an electric dipole moment in another frame when the source is in motion. This interplay underscores the unified nature of electromagnetism in the theory of special relativity and is crucial for understanding phenomena where moving charged particles, like electrons, display characteristics of both magnetic and induced electric dipoles.

Transcript

00:01 Hi, in the given problem there is an electric dipole kept vertically like this.

00:11 The two charges positive plus q and negative minus q.

00:25 These two charges are kept as shown in the figure and observation point is at the equatorial line.

00:37 Here, this observation point is at a distance, this is named as p, and this is at a distance from the center of this dipole r, and this distance r is quite large as compared with the gap between the two charges of this dipole, means the dipole length.

01:02 So this r is very much larger than this d.

01:06 Now we have to find net elective field at this observation point p, for which, first of all, we join the charges with the observation point and find their distances.

01:23 So as the observation point p is at equator of the dipole.

01:47 So we can say distance of observation point from both of the charges should be same.

01:54 So if we consider this plus q to be put at a point a minus q to be put at a point b, then we can say this ap is equal to b .p.

02:05 And this will be d by 2 as square root of r square plus d by 2 whole square means d square plus d square by 4.

02:22 Now as the distances are same, so these two angles will also be same, theta h.

02:28 And now the two electric fields at this observation point, as we know, electric field always goes away from the positive charge.

02:35 So due to plus q put at a, it will be going away like this.

02:41 And due to minus q put at b, it will be approaching the negative charge like this.

02:46 So these are the two electric fields.

02:51 Ea and e b.

02:54 Now as the magnitude of charges is same and the distances ap and bp are also same...

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