In a Hall-effect experiment, a current of 3.0 A sent lengthwise through a conductor 1.0 cm wid

In a Hall-effect experiment, a current of 3.0 A sent...

In a Hall-effect experiment, a current of $3.0 \mathrm{~A}$ sent lengthwise through a conductor $1.0 \mathrm{~cm}$ wide, $4.0 \mathrm{~cm}$ long, and $10 \mu \mathrm{m}$ thick produces a transverse (across the width) Hall potential difference of $10 \mu \mathrm{V}$ when a magnetic field of $1.5 \mathrm{~T}$ is passed perpendicularly through the thickness of the conductor. From these data, find (a) the drift velocity of the charge carriers and (b) the number density of charge carriers. (c) Show on a diagram the polarity of the Hall potential difference with assumed current and magnetic field directions, assuming also that the charge carriers are electrons.

Key Concepts

Lorentz Force

The Lorentz force is the force exerted on a charged particle moving through electric and magnetic fields, given by the equation F = q(E + v × B). In the context of the Hall effect, the magnetic part of the Lorentz force deflects the moving charge carriers, leading to the accumulation of charges on one side of the conductor and the creation of a potential difference. This concept is key to understanding how external magnetic fields influence the motion of charge carriers in conductors.

Charge Carrier Number Density

The number density of charge carriers describes the concentration of mobile charges per unit volume within a material. This parameter is crucial for analyzing the electrical conductivity and response of materials to electric and magnetic fields. In experiments like the Hall effect, knowing the number density helps in relating the measurable Hall voltage and current to the fundamental properties of the charge carriers, such as their drift velocity.

Hall Effect

The Hall effect is a phenomenon in which a voltage difference (the Hall voltage) is generated across an electrical conductor, transverse to the direction of the current and an applied magnetic field. This happens because the magnetic field exerts a Lorentz force on the moving charge carriers, causing them to accumulate on one side of the conductor, thereby creating a measurable potential difference. Understanding the Hall effect is fundamental in studying charge carrier properties and the behavior of conductive materials in magnetic fields.

Drift Velocity

Drift velocity is the average velocity at which charge carriers, such as electrons, move through a conductor under the influence of an electric field. It is a key concept in electrical conduction because it links the microscopic motion of carriers to the macroscopic current. The drift velocity is typically very small compared to the random thermal velocities of the carriers, but it directly determines the current when combined with the carrier density and charge.

Transcript

00:01 Crossed or perpendicular electric and magnetic fields are often used in the study of how charges behave in materials.

00:10 Here's another example that's often used in solid state physics called the hall effect.

00:17 It's a small effect, so you do need a fairly large magnetic field and a fairly sensitive volt meter.

00:26 Here i show a current coming into a slab of material.

00:31 From the left, traveling to the right.

00:34 It's a fairly large current.

00:36 If we look at putting this slab in a magnetic field as shown, pointing up through the face of the slab, then what will happen to the charge carriers if they are positive is they will be deflected to the bottom end of the slab, leaving it positive relative to the other end.

01:05 And a potential difference will develop between the two sides.

01:13 What's even more interesting is if it is electrons that are the charge carriers, which it typically is for metals.

01:22 Let's see what happens there.

01:25 They will have a velocity opposite the current, and then they will be forced to the bottom side.

01:37 And the positive charges will scoot to the.

01:40 The high side, top side in my drawing.

01:49 And so you will get a different polarity of voltage depending on which type of charge, positive or negative, is actually carrying the current.

02:02 So yes, there is a difference.

02:05 The potential difference that you measure across the two sides is called the hall potential, or vh will call it.

02:17 All potential.

02:24 And we are going to look at the situation where it is 10 microvolts.

02:32 Remember, micro means 10 to the minus 6.

02:35 So yes, these are, it's a fairly small effect.

02:39 You need a strong magnetic field, a large current, and a sensitive volt meter in order to detect it.

02:47 But in the steady state situation after you've turned on power supplies to the magnetic field in the current, what you have happening is there's a whole potential that makes an electric field between the two sides.

03:07 We'll call that eh equals v hall divided by the width in this case.

03:14 And the the electric force on the electrons is canceled by the magnetic force.

03:29 And that will happen at a certain speed, as we can see by putting in qe for the electric force, e hall in this case, and that is equal to qe times b.

03:46 And so the idea is, what's this electric field is set up, the electron will continue to move straight through the material with a drift velocity.

04:00 And so you can get that drift velocity, of course the charge cancels, so you don't even need to know what that is for the charge carriers.

04:09 And the drift velocity then can be measured from the hall electric field, which comes from the measurement of the whole potential, and the width of the sample over which the the potential develops and the external magnetic field.

04:32 So this is the drift speed and the direction of the velocity you get from the carriers, are they positive or negative? okay, so in this case, let's take a look at this.

04:50 The whole potential width was one centimeter...

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