Measurement of Voltages, Currents and Resistances

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Voltmeter and Ammeter

These devices measure the voltage and current respectively in a circuit. The basic component of both is the moving coil galvanometer which produces a deflection proportional to the electric current through it.

Ammeter

An ammeter is connected in series with the circuit element whose current is to be measured, so that there is only a negligible change in the circuit resistance and hence circuit current.

Let the galvanometer resistance be G and the current for full-scale deflection be I_g. To measure larger currents, a suitable low resistance S (called shunt) is connected in parallel with the galvanometer.

The value of S is chosen by the maximum current I that we want to measure. This means that though the circuit current is I, only a current I_g should be through the galvanometer. The remaining current

I - I_g = I_s should flow through the shunt. Equating potential differences across the shunt and galvanometer, we get

(I - I_g) S = I_g G

The resistance of the ammeter (i.e., shunted galvanometer) is

\ R_A < S

So, the shunt not only extends the range of current (from I_g to I), it extends the range of current (from G to R_A) of the ammeter.

Voltmeter

A voltmeter is connected in parallel with the circuit element across which potential difference is to be measured. It should have a very high resistance as not to alter the circuit resistance, and hence circuit current.

The galvanometer can measure voltages upto I_GG. For larger potential differences, a suitable high resistance R (called multiplier) is connected in series.

The value of R is chosen according to the maximum voltage V that we want to measure. But the galvanometer by itself can only handle a voltage of I_gG. The remaining potential difference (V - I_gG) should be across the multiplier R. The current through it is I_g. Therefore, equating voltage drops, we get

V = I_g G + I_g R

The resistance of the voltmeter (i.e., a galvanometer in series with a high resistance) is

R_V = G + R

Since R is high, the multiplier increases the resistance of the voltmeter, and of course, extends the voltage range (from I_gG to V).

Wheatstone Bridge

This is used to measure an unknown resistance accurately. It consists of 4 resistors (2 fixed known resistances P and Q, a known variable resistance R and the unknown resistance X) connected as shown in the figure.

Wheatstone's network

A source of emf is connected across one pair of opposite junctions (A and C), and a galvanometer G across the other opposite pair (B and D). The key K₁ is closed first and then K₂. The value of R is varied till the galvanometer shows no deflection, i.e., I_g = 0. Then, the bridge is said to be balanced.

The wheatstone bridge principle states that under balanced conditions, the products of the resistances in the opposite arms are equal, i.e.,

We will show this using Kirchoff's rules.

Applying the Kirchhoff's law to loop 1, we have

-I₁P - I_gR_g + (I - I₁)R = 0 ...(1)

Similarly for loop 2, we have

- (I₁-I_g)Q +(I - I₁+ I_g)X +I_gR_g = 0 ...(2)

(where R_g is the resistance of the galvanometer)

In the balanced condition, putting I_g = 0, we have

-I₁P + (I - I₁)R = 0 …(1)

and -(I₁)Q + (I - I₁)X = 0 …(2)

Simplifying the two equations, we get

I₁P = (I-I₁)R …(1)

I₁Q = (I-I₁)X …(2)

Dividing the above two equations, we get

Resistor Q is called the standard arm of the bridge, and resistor P and R are called the ratio arms.