3.2 The Wheatstone bridge

Originally developed in the nineteenth century, a Wheatstone bridge provided an accurate way of measuring resistances without being able to measure current or voltage values, but only being able to detect the presence or absence of a current. A simple galvanometer, as illustrated in Figure 13, could show the absence of a current through the Wheatstone bridge in either direction. The long needle visible in the centre of the galvanometer would deflect to one side or the other if any current was detected, but show no deflection in the absence of a current.

Figure 13  An early galvanometer showing magnet and rotating coil

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This is a photograph of an early galvanometer. A long needle extends from the centre to a curved measurement scale, allowing the deflection of the needle in either direction to be measured.

Figure 13  An early galvanometer showing magnet and rotating coil

Figure 14(a) shows a circuit made of four resistors forming a Wheatstone bridge. Its purpose here is to show whether there is any current flowing between and . Figure 14(b) shows an equivalent way of drawing the circuit.

Figure 14  Equivalent examples of a Wheatstone bridge

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This is a Wheatstone bridge with four resistors in a diamond configuration next to an equivalent version with the resistors arranged in a vertical rectangle. In each case, resistors R₁ and R₂ are connected in parallel to resistors R₃ and R₄. The voltage between R₁ and R₂ is denoted V_left and the voltage between R₃ and R₄ is denoted V_right. These two points are connected across the centre of the bridge via an ammeter.

Figure 14  Equivalent examples of a Wheatstone bridge

The bridge is said to be balanced (that is, no current flows through the bridge and the needle of the galvanometer shows no deflection) if the voltages and are equal. It can be shown that the bridge is balanced if, and only if, , as follows.

When then . Then the Wheatstone bridge can be viewed as two voltage dividers, and on the left and and on the right. Applying the voltage divider equation gives and .

So

and

Multiplying out the brackets gives

which simplifies to

and

So, if were unknown, , and could be chosen so that the needle of a galvanometer showed no deflection due to the current. Then