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Ohm's Law

Teacher/Instructor Jonathan Osbourne
Jonathan Osbourne

PhD., University of Maryland
Published author

Jonathan is a published author and recently completed a book on physics and applied mathematics.

Ohm's law defines a concrete relationship between potential difference and current flow. This relationship is satisfied by all ohmic materials according to the equation potential difference = - current x resistance.

So let's talk about Ohm's law. What is Ohm's law? Well, Ohm's law gives us a concrete relationship between potential difference and current flow. So, how does it do this? Well, we've got a basic relationship. It's satisfied by all materials that are so called ohmic materials and this relationship tells us that the current density which is current divided by area is equal to the conductivity which depends only on the material. So copper has a conductivity, silver has a conductivity, whatever, times the electric field. So when I apply an electric field, I multiply by the conductivity and that tells me how much current flows.

Now, a good conductor has a really really really big big conductivity and that means that even a tiny electric field, will make lots of current density flow. Alright. Now, let's try to turn this into something that's associated with the current rather than the current density. And something that's associated with the potential difference rather than the electric field because these are the things we have easy access to measure, current and potential difference. We don't want to measure current density and electric field because they're more annoying. Alright.

So, let's take a wire like this. He's got a cross sectional area a and we're going to apply an electric field across him. Now we're going to consider a length l of this wire. Okay, the potential difference across this wire is given by negative electric field times length. And that's just like the work relationship. Work equal force dot displacement and then potential energy or change in potential energy equals minus the work. So that's where this minus sign comes from. Alright.

So, we can solve this for the electric field and then plug that into this ohmic relationship. And that gives us j equals minus sigma times delta v over l. Now we're going to solve this for delta v and that will give us delta v equals minus j divided by sigma times l. Alright. Now, remember that the current density was current divided by area. So if I plug that in, I'll get minus current and then times this little one over sigma times the length of the wire divided by the cross sectional area. Alright, that looks annoying. So we're going to give it a name. We're going to call it r, the resistance. Alright. Let's go on over here.

So the resistance is equal to one over sigma, one over the conductivity, that's called the resistivity, alright, times the length of wire divided by the cross sectional area. Now the nice thing about this is that this resistivity depends only on the material. It doesn't depend on how big the wire is or how long it is or all the geometrical properties of the wire. It only depends on what it's made of. Alright. So what that means is that if I've got a thick wire with a large cross sectional area, then that will give me a really small resistance. If I've got a long wire, really long, then that will give me a large resistance. Okay. We can again think about this in terms of traffic. Thick wire, that's like many lanes of traffic, so that's going to be easier for people to drive down in. Long wire, that's like really long road, less people are going to choose that option. Alright.

So let's do some problems. Oh, we've got delta v equals minus i r. This is what people usually are referring to when they say Ohm's law. Change of the potential difference equals minus i r. So what does this minus sign mean? Well the minus sign means that the current flows in the direction of decreasing potential. So if I go across a resistor, in the direction of the current, then that means that the potential is decreasing. So the change of potential will be negative. Alright. Let's do some problem.

So, what is the potential difference across a 3 ohm resistor that carries 4 amps of current? Alright. Ohm's law directly. Potential difference equals minus ir equal minus, what's the current? 4 amps. What's the resistance? 3 Ohm's. So it will be 12 volts. Notice I don't have to worry about the units because everything's in SI units. So that means, it's a potential difference, it's volts. Done. Alright.

Determine the current in a 5 ohm resistor with a 30 volt potential difference maintained across it. Alright. Now I want the current. So I'm going to start again with Ohm's law. Delta v equal i r. I'm not worried about the minus sign, well, I'll put it there. But it doesn't really mater as far as what this is concerned with. Alright. So we'll say 30 volts equals minus i times resistance, 5 Ohm's. So that means the current must be equal to 6 amps. Again the minus sign's kind of annoying, really I should have a minus sign there but whatever. I just want to know what's the current, right? The minus sign tells me what direction the current's going in. And it doesn't ask me that. Alright.

Number 3. What is the resistance of a resistor that carries a current of 18 amps when a 36 volt potential difference is maintained across it or imposed across it. Alright. Once again, Ohm's law. Delta V equal i r. Well, I'm just going to ignore the minus sign this time.

Alright. Delta v is 36, i 18 r. So I'll divide and the resistance will be 2 Ohm's. Notice that I even left the units out this time. Basically, you really need to give units in the answer, whether or not you give units when you're doing the Algebra, that depends on your teacher. Some teachers want you to write units everywhere and if your teacher wants you to do it, you got to do that. But the important thing is to make sure that you always have a unit in your answer.

Alright. That's Ohm's law.