In the Bohr model of the hydrogen atom, the electron moves...

In the Bohr model of the hydrogen atom, the electron moves in a circular orbit of radius $5.3 \times 10^{-11} \mathrm{~m}$ with a speed of $2.2 \times 10^{6} \mathrm{~m} / \mathrm{s}$. If we are viewing the atom in such a way that the electron's orbit is in the plane of the paper with the electron moving clockwise, find the magnitude and direction of the electric and magnetic fields that the electron produces at the location of the nucleus (treated as a point).

Key Concepts

Right-Hand Rule

The right-hand rule is a mnemonic tool used to determine the direction of the magnetic field around a current or moving charge. In the context of a current loop, pointing the thumb in the direction of the conventional current (which is opposite to the motion of a negative charge) will indicate the orientation of the generated magnetic moment and therefore the direction of the magnetic field at key points.

Biot–Savart Law

The Biot–Savart law is a fundamental equation that relates current elements or moving charges to the magnetic field they produce. For a current loop, the law simplifies to a form that directly connects the magnetic field at the center of the loop to the current and the radius of the loop, capturing how a circulating current generates a dipole magnetic field.

Current Loop Equivalence

When a charge moves along a closed path repetitively, its motion can be equivalently described as a steady current loop. This equivalence allows one to apply formulas for the magnetic field of a current loop, such as the expression for the magnetic field at the center of a circular loop, which is useful in analyzing systems with periodic motion.

Electric Field of a Point Charge

This concept explains that any point charge creates an electric field in the space around it, the magnitude and direction of which are given by Coulomb’s law. The field strength is proportional to the magnitude of the charge and varies inversely with the square of the distance from the charge. The direction of the field is radially outward from a positive charge and radially inward toward a negative charge.

Magnetic Field from a Moving Charge

A moving charge produces a magnetic field, which can be calculated using principles from classical electrodynamics such as the Biot–Savart law. The magnetic field produced by a moving charge depends on both the speed of the charge and its direction of motion, and its vector direction is determined by the cross-product between the velocity of the charge and the position vector.

Transcript

00:01 Problem 28 .2, we have the bore model of hydrogen atom, which the electron moves in a circular orbit of radius 5 .3 times 10 to the negative 11 meters.

00:11 That's this vector here, r, with the speed of 2 .2 times 10 to 6 meters per second.

00:17 That's this vector b.

00:18 And we're reviewing the atom in such a way that electrons over in the plane of paper with the electron moving clockwise.

00:24 We find the magnitude and direction of the electric and magnetic fields at the point produced by the particle.

00:31 Okay, so let's start with the magnetic field, magnetic field.

00:36 So we start with the equation for the magnet field of a moving point charge.

00:41 So that's going to be b is equal to mu not over 4 pi.

00:50 And then we have qv, sine, phi over r squared.

01:02 Okay, so what we need to do is we need to first write in what the charge of the lauchan is.

01:11 So that's just going to write down as qe.

01:14 So mu not over 4 pi.

01:19 And then we write down what v is.

01:21 That's just a number that we're given.

01:23 And then sine of phi is something we need to figure out.

01:25 So sine of phi, that phi represents the angle between the vector b and r.

01:31 And so since that is negative degrees, we have sine of pi over two, which comes out to be one.

01:45 Okay? and we also have v in here, slash this number again...

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