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Show that the Hall voltage across wires made of the same material, carrying identical currents, and subjected to the same magnetic field is inversely proportional to their diameters. (Hint: Consider how drift velocity depends on wire diameter.)

$\varepsilon \propto \frac{1}{l}$

Physics 102 Electricity and Magnetism

Chapter 22

Magnetism

Magnetic Field and Magnetic Forces

Sources of Magnetic field

Electromagnetic Induction

Inductance

University of Washington

Hope College

University of Sheffield

McMaster University

Lectures

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hi, everyone. So we need to. I need to find a relation between a whole voltage and the wire's diameter. So imagine we have two identical, not identical. Two wires need from same material and magnetic fields are the same. And Kuerten's reality Kuerten's in through these wires are also the same on Lee. Difference is in diameters. Uh, and we want to show that Walt Ege Acre Wilton May measured across the wire, is proper inversely proportional to its diameter. So let's get started. They're in. Consider were considered a wire inside the wire. There now are many other chance Seems no charge carrier. He's charge carriers are eloquence, Indication from NATO. And we have a magnetic field here, and there will be created an electric field which would can't compensate the magnetic force. So cute we help, um, plus charges and minus judges on the sites. Does Electric field is looking downwards. And while let's let's consider one of this electron and magnet e forces e b V, where e he's, uh, Electron's charge and Ellen Creek Force would be E capital E, which is electric field and for and equally room, they should be equal. And when we can. So the car keys we all lived. We'd this relation and we can find the u Let's call it U World Ege. He's generally e multiplied by D They're de ese diameter. So let me show This is D and this is simply v p. D. And we have a schemed here that Yeah, it says, um, look, Att look att The drift velocity, uh, and how it is depended on diameter. So we know that Hee hee is constant in both cases, it doesn't change and that let's write down the formula for E electric Current is an e iss the where I will explain every term. So in is number there's density off electrons. So we chose how many eloquence are located in one could be commit meter and this is constant. Ah, but this is property off a specific metal and scenes. In both cases, the same metal is used and doesn't change anything. The same for Brussels thing. Mitchell. Both of the wires e He's just an electron's charge and S is a cross sectional area. And these three fellow city, which we hope here in our formula and, uh, voltage and we can rewrite this so except s and V here everything is constant is envied. And we also know that is if the curse curse. Ah, section is circular. Ah, and let's go Considered circ circ circle. Ah, jeez. E t squared over for it is unknown. Formal off, huh? Yeah, an area Officer Cho. And then we have we And this should be i o r and e And look a tte the rights. Ah, right inside everything is constant for for the both wires. So we can say that we d squared is constant. 40 bullet, both wires. And, uh, then if we continue this equation and forget about P as well, because he's still, uh, still constant So be the same for both. Why years and we have from these formula directly we have if we owe me to be and right down um proportion you is proportional toe the a multiplied by D And if we use our burn result that weise precaution all in a very sleep proportional to he squared and plug in here we would get instead of we we have ah wonder where d squared. Um Seems we have proportional proportions signed not equal sign here, we can simply ah, simply switch. We live one over D squared, and we didn't We don't wile it anything. We wouldn't violate anything, so Ah, And one these can sold and we're left with won over deep and disease. What was asked, um, we needed to this relation, and we did.

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