A potentiometer is a device to precisely measure potential dif...

Question

A potentiometer is a device to precisely measure potential differences or emf, using a “null” technique. In the simple potentiometer circuit shown in Fig. 26–83, R′ represents the total resistance of the resistor from A to B (which could be a long uniform “slide” wire), whereas R represents the resistance of only the part from A to the movable contact at C. When the unknown emf to be measured, Eₓ , is placed into the circuit as shown, the movable contact C is moved until the galvanometer G gives a null reading (i.e., zero) when the switch S is closed. The resistance between A and C for this situation we call Rₓ . Next, a standard emf, Eₛ, which is known precisely, is inserted into the circuit in place of Eₓ and again the contact C is moved until zero current flows through the galvanometer when the switch S is closed. The resistance between A and C now is called Rₛ. Show that the unknown emf is given by

Eₓ = (Rₓ / Rₛ) Eₛ

where Rₓ, Rₛ and Eₛ are all precisely known. The working battery (at top of circuit diagram), as well as the standard cell Eₛ, are assumed to be fresh and to give a constant voltage.

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Eₓ = (Rₓ / Rₛ) Eₛ

where Rₓ, Rₛ and Eₛ are all precisely known. The working battery (at top of circuit diagram), as well as the standard cell Eₛ, are assumed to be fresh and to give a constant voltage.

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem to measure a known reference voltage V zero. A potent meer setup is used by comparing it with an unknown voltage V connected between points X and Y. A long uniform resistance wire with a total resistance of RT includes a movable contact point P that slides along the wire to adjust the resistance between X and P. First, the circuit is connected to the known voltage V zero and P is adjusted until a null reading zero current appears on the Gallo Meer, G at this null position. The resistance between X and P is R zero. Next, the circuit is switched to the unknown voltage V and P is adjusted again until the Gallo Meer shows a null reading. Add this new null position. The resistance between X and P is RV, determine an expression for the unknown voltage V in terms of V zero RV and R zero. Awesome. So it appears our final answer that we're ultimately trying to solve for our end goal of sorts is we're trying to determine an expression to help us solve for the unknown voltage V. So we're ultimately trying to figure out the expression for the unknown voltage V using the terms V zero RV and R zero. Awesome. So now that we know that we're trying to figure out an expression for the unknown voltage V. Let's take a moment here to briefly go over the circuit that's provided to us. So as we could see, we have a movable contact point P denoted on our resistor, which we can know it's a resistor, but it's a jagged line that looks like a lightning bolt. And we have a giant long resistor from point X to Y. Then we also have a smaller resistor RV at the top of our circuit that is right next to our working battery. And we know that that's a battery because as we should recall, when we're dealing with the circuit diagram, two parallel lines where one line is slightly longer than the other denotes the battery. And then looking at our long resistor that's from point X to point Y when we have our contact point P. So from the distance from point X to contact point P is some distance R apart and from point X to point Y is some distance RT apart denoted by dotted arrows. And then we have our voltage which is V zero or V. And then we have a Gallo Meer denoted by a circle with a capital G inside of it. Awesome. And that is it, that's pretty much all the major points of our circuit here. So now let's read off our multiple choice answers to see what our final answer might be for our expression. And note that it says V equals sum expression. So A is R zero divided by RV, multiplied by V zero. B is RV divided by R zero multiplied by V zero. C is R zero plus RV divided by R zero, all multiplied by V zero. And finally, D is RV minus R zero, all divided by R zero multiplied by V zero. OK. So it appears that our first step for solving for this particular problem is we need to consider the null condition. So as we should recall and note in both cases for the known voltage V zero and the unknown voltage va null reading on the gallon meer means that there is no current flowing through it. Therefore, this indicates that the potential difference across the segment of wire from point X to P exactly matches the same voltage, either V zero or V depending on which is connected. Awesome. So with that in mind, we now need to recall and set up the equations that are required to help us solve for each voltage. So with that in mind, let us start with the known voltage V zero. So as we should recall a note at the null point for V zero, the voltage across the segment X to P is equal to V zero. And since the total current denoted by I flowing through the wire remains constant due to the fact that the staple battery is working across RT, which once again, RT is our total resistance. We can express our voltage equation for V zero as follows. So V zero is equal to the current I multiplied by R zero. Fantastic where R zero in this case is the resistance between X and P and at the null point for V zero. So once again, R zero is the resistance between X and P at the null point for V zero. So now moving right along here, we are now gonna consider the unknown voltage V. So as we should recall, similarly, at the null point for V, the voltage across the segment X to P is also equal to V and we can write this equation as V equals the current I multiplied by RV. Where RV, in this case is the resistance between X and P at the null point V. Awesome. So now our final step is we need to equate the currents and we can equate the currents because as we should recall, since the current eye through the point Teetter wire, po sorry, the potent meer. So the potent meer wire is the same. So the current that's flowing through this wire is the same for both cases. And because of that reason, we can therefore equate the two expressions for the current I, meaning we could go ahead and write that V zero divided by R zero is equal to V divided by RV. And now we just need to rearrange this expression to isolate and solve for V because V is the final answer we're ultimately trying to solve for. So we use a little bit of algebra, we will find that V is equal to RV, divided by R zero, all multiplied by V zero. And that's it we've solved for this problem. Hooray, we did it. So when we look at our multiple choice answers, the correct answer has to be the multiple choice answer letter B which states that V is equal to RV divided by R zero all multiplied by V zero.

Key Concepts

Potentiometer Principle

Ohm's Law

Null Method

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