# Lesson Explainer: Design of the Ohmmeter Physics

In this explainer, we will learn how to describe the combining of a galvanometer with fixed and variable resistors to design a dc ohmmeter.

An ohmmeter is a device used to measure the resistance of a resistor. Using Ohm’s law, we know we can find the resistance of a circuit by dividing the voltage by the current:

Let’s consider how we can find these values by attaching a galvanometer to a circuit with a resistor in it.

Recall that galvanometers measure current. When looking at the galvanometer’s needle, a larger deflection means a larger current, which means a smaller resistance. A smaller deflection means a larger resistance.

If the meter shows zero current, this means the resistance of the circuit is infinite, so it must always be deflecting even if just slightly. For the purpose of seeing if deflection occurs, the galvanometer is typically modified to have it start at zero rather than at negative values of current.

The maximum value of the current that can be measured by the galvanometer is . Every point on the dial is thus a fraction of the current: half deflection means a current of , quarter deflection is , and so on.

We ideally want to add a certain amount of resistance to the galvanometer, such that the total resistances make the current . We do this by putting in two different resistors: a fixed resistor, , and a variable resistor, . This, combined with the galvanometer’s own resistance , makes up all the resistances in the circuit before anything is added.

These resistances are such that when there is no other resistance in the circuit, the current is exactly enough to cause maximum deflection of the meter, . What this means is that we want to relate the total resistance of this constructed ohmmeter, , to as follows: where the total resistance of the ohmmeter is

We can adjust the variable resistor to a specific value, the exact value being related to the other values. To show this, we must start with the Ohm’s law relation with the total ohmmeter resistance:

We can replace with its constituents as follows:

Now we can solve for by subtracting and from both sides:

Let’s look at an example using this equation.

### Example 1: Variable Resistance Adjustment of Ohmmeter

A circuit that can be used as an ohmmeter is shown. The circuit uses a galvanometer with a resistance of 50 Ω that has a full-scale deflection current of 0.5 mA. The circuit also includes a direct current source with a voltage of 3.5 V, a fixed resistor with a resistance of 2.5 kΩ, and a variable resistor. The resistance of the variable resistor is adjusted until the galvanometer arm is at a full-scale deflection position. What resistance is the variable resistor set to? Answer to the nearest ohm.

### Answer

The resistor on the left represents the variable resistor.

Let’s look at the equation

The voltage is 3.5 V and the resistance of the galvanometer, , is 50 Ω. For the other values, we’ll need to convert to the base units first. There are 1‎ ‎000 Ω in 1 kΩ: so

Therefore, the fixed resistor is 2‎ ‎500 Ω.

Now let’s look at the amperes. There are 1‎ ‎000 mA in 1 A: so

The full-scale deflection for the current, , is 0.0005 amperes.

Thus, we can put these values into the equation for the variable resistor:

Volts over amperes gives units of ohms, giving a variable resistance value of

To the nearest ohm, the value of this variable resistor should thus be set to 4‎ ‎450 Ω.

Now that we know how to set up the ohmmeter, let’s look back at how the galvanometer’s scale is calculated more exactly. If the meter starts at 0 current and goes all the way up to , then we can relate the current to the total resistance in the circuit like this:

This means that the dial on the galvanometer starts at an infinite resistance, then deflects until it reaches 0 at the far end.

Combined with the resistors, this creates our ohmmeter. To measure an unknown resistor, it must be in series with the other parts of the ohmmeter.

Let’s look at an example.

### Example 2: How an Unknown Resistor Must Be Connected

A circuit that can be used as an ohmmeter is shown. The circuit uses a galvanometer, a direct current source with a known voltage, a fixed resistor, and a variable resistor. The resistance of the variable resistor is adjusted until the galvanometer’s arm is at a full-scale deflection position. The circuit is to be used to find the resistance of a resistor that has an unknown resistance. The resistor with the unknown resistance must be connected to the circuit. In which of the following ways should the resistor be connected?

1. In series with all the other components
2. In parallel with the variable resistor
3. In parallel with the direct-current source
4. In parallel with the fixed resistor
5. In parallel with the galvanometer

### Answer

If a component, whether it be a galvanometer, a direct-current source, or a variable or fixed resistor, is in parallel, it will not measure correctly. The only way the ohmmeter, with its variable resistor adjusted to have a full deflection, will measure the unknown resistor correctly is by putting the unknown resistor in series.

The correct answer is A.

The way that we measure the resistance using this ohmmeter is by proportion of the galvanometer needle’s deflection, with a larger deflection meaning a larger current and thus a smaller resistance.

Let’s look at an example.

### Example 3: Determining Resistance Using Specific Needle Deflection

A circuit that can be used as an ohmmeter is shown. The circuit uses a galvanometer, a direct current source with a known voltage, a fixed resistor, and a variable resistor. The angle is the full-scale deflection angle of the galvanometer. Two resistors, and , are connected to the ohmmeter so that their resistances can be measured by the ohmmeter. The galvanometer’s angle of deflection is reduced by the angle when is connected, and its angle of deflection is reduced by the angle when is connected; . Which of the following correctly relates the resistances of and ?

### Answer

A larger resistance would show on a meter as a smaller relative deflection, and a smaller resistance as a larger deflection. These angles the needles are deflecting are from the same ohmmeter calibrated the same way, so since there is a difference between them, cannot equal . So A is not it.

The angle with the largest deflection will thus have the smallest resistance. Since , and represents , is a smaller value of resistance than . So the correct answer is C, .

If an unknown resistor is added to the circuit and the needle deflects halfway, this means the added resistance is equal to the resistance in the ohmmeter, . Thus, we can update the resistance markings on the galvanometer to show this relation in the center.

The deflection is inversely proportional to the resistance. This position halfway between the center and the right of the meter is , and the one between the center and the left is , and so on.

is halfway between the point on the left and the point of , is halfway between there and , and so on. Each additional ohm provides a smaller and smaller needle deflection as it gets closer to the infinity side. The needle deflection is nonlinear. To see the exact relation, let’s look at the proportion of the deflection.

Let’s call the needle’s deflection proportion , which is the proportion of current deflection over maximum deflection:

The deflection of the needle is typically measured in degrees, which becomes just a proportion when divided this way. This deflection can then be related to the actual value of the unknown resistance by comparing it to the ohmmeter resistance.

If we divide the resistance of the ohmmeter, , by this proportion and then subtract the ohmmeter resistance, this gives the value of the unknown resistor:

Let’s look at some examples.

### Example 4: Determining Resistance Using Halfway Needle Deflection

The diagram shows the scale of an ohmmeter that is being used to measure an unknown resistance. The resistance of the ohmmeter is 25 kΩ. The angle of full-scale deflection of the ohmmeter . The angle of deflection of the ohmmeter arm . What is the unknown resistance? Answer to the nearest kilohm.

### Answer

The full-scale deflection of this ohmmeter at is 0 Ω, and at .

The halfway point in between is equal to the resistance of the ohmmeter, which we are given. So the unknown resistance is simply 25 kΩ, but let’s also find this by using the equation.

The deflection proportion is 0.5, so we put these values into the equation to get

So, the unknown resistance is 25 kΩ.

### Example 5: Determining Resistance Using Specific Needle Deflection

The diagram shows the scale of an ohmmeter that is being used to measure an unknown resistance. The resistance of the ohmmeter is 25 kΩ. The angle of full-scale deflection of the ohmmeter . The angle of deflection of the ohmmeter arm . What is the unknown resistance? Answer to the nearest kilohm.

### Answer

The full-scale deflection of this ohmmeter at is 0 Ω, and at . The ohmmeter resistance, , is 25 kΩ

The proportion of deflection is found by taking the degree of the deflection over the total deflection, so in this case, 48 degrees over 60:

Using this proportion in the equation,

To the nearest kilohm, the unknown resistance is thus 6 kΩ.

### Example 6: Determining Resistance Using Small Needle Deflection

The diagram shows the scale of an ohmmeter that is being used to measure an unknown resistance. The resistance of the ohmmeter is 25 kΩ. The angle of full-scale deflection of the ohmmeter . The angle of deflection of the ohmmeter arm . What is the unknown resistance? Answer to the nearest kilohm.

### Answer

The full-scale deflection of this ohmmeter at is 0 Ω, and at . The ohmmeter resistance, , is 25 kΩ

The proportion of deflection is found by taking the degree of the deflection over the total deflection, so in this case, 6 degrees over 60:

Using this proportion in the equation,

To the nearest kilohm, the unknown resistance is thus 225 kΩ.

Let’s summarize what we have learned in this explainer.

### Key Points

• An ohmmeter can be created by putting a fixed resistor, a variable resistor, and galvanometer in series with one another.
• To calibrate the galvanometer, the fixed and variable resistor values must be chosen such that the current is equal to the full-scale deflection of the galvanometer.
• Ohmmeters have a nonlinear scale of needle deflection from to 0 Ω.

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