# 13.8.2 More Complicated Circuits

Protection Resistor

In practice, the potentiometer circuit can be more complicated. For example, a protection resistor R2 may be added in series with E2. This is to limit the magnitude of the current i while we are still searching for the null deflection. However, having R2 does not complicate our calculation of E2 at all. This is because at null deflection, VCD is still equal to E2. This is because at null deflection, no current is flowing through R2 so the PD across R2 is zero.

Protection Resistor

Similarly, the internal resistance of E2 does not complicate our measurement of E2 at all. Again, this is because at null deflection, VCD is still equal to E2. This is because at null deflection, no current is flowing through r so the PD across r is zero.

Precision Resistor

We also often include a resistor R1 in the driver circuit. By the Potential Divider Principle, if the total resistance of the slide wire is Rw, then $\displaystyle {{V}_{{AB}}}=\frac{{{{R}_{w}}}}{{{{R}_{w}}+{{R}_{1}}}}{{E}_{1}}$

The calculation of E2 is now going to be \displaystyle \begin{aligned}{{V}_{{CD}}}&={{V}_{{AX}}}\\{{E}_{2}}&=\frac{{AX}}{{AB}}{{V}_{{AB}}}\\&=\frac{{AX}}{{AB}}\frac{{{{R}_{w}}}}{{{{R}_{w}}+{{R}_{1}}}}{{E}_{1}}\end{aligned}

Having R1 reduces VAB. While this decreases the range of voltages the potentiometer can balance, it has the benefit of a longer balance length AX. A smaller percentage uncertainty in the measurement of AX ultimately results in a more precise measurement of E2.

The diagram below shows a potentiometer being used to measure not just the EMF, but terminal PD Vt of cell E2 at different loading. (Can’t they just use the voltmeter? Sigh)

Notice E2 now has a complete circuit of its own called the secondary circuit. So there is a current I3 running through the internal resistance r even at null deflection. This means that VCD is no longer equal to E2 even at null deflection (unless R3 is infinitely large)

However, since there is no current flowing between driver and secondary circuit at null deflection, we can still analyse the two circuits separately, as if they are not connected to each other.

For the driver circuit, we have \displaystyle \begin{aligned}{{V}_{{AX}}}&=\frac{{AX}}{{AB}}{{V}_{{AB}}}\\&=\frac{{AX}}{{AB}}\frac{{{{R}_{w}}}}{{{{R}_{w}}+{{R}_{1}}}}{{E}_{1}}\end{aligned}

For the secondary circuit, we have $\displaystyle {{V}_{{CD}}}={{V}_{t}}=\frac{{{{R}_{3}}}}{{{{R}_{3}}+r}}{{E}_{2}}$

VCD is still equal to VAB, so E2 can be calculated through \displaystyle \begin{aligned}{{V}_{{CD}}}&={{V}_{{AX}}}\\\frac{{{{R}_{3}}}}{{{{R}_{3}}+r}}{{E}_{2}}&=\frac{{AX}}{{AB}}\frac{{{{R}_{w}}}}{{{{R}_{w}}+{{R}_{1}}}}{{E}_{1}}\end{aligned}

Video Explanation