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, V_CD 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, V_CD 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 R_w, 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 V_AB. 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.

Adding a Secondary Circuit

The diagram below shows a potentiometer being used to measure not just the EMF, but terminal PD V_t 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 V_CD 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

$\displaystyle {{V}_{{CD}}}={{V}_{t}}=\frac{{{{R}_{3}}}}{{{{R}_{3}}+r}}{{E}_{2}}$

V_CD is still equal to V_AB, so E₂ can be calculated through

$\displaystyle \begin{aligned}{{V}_{{CD}}}&={{V}_{{AX}}}\\\frac{{{{R}_{3}}}}{{{{R}_{3}}+r}}{{E}_{2}}&=\frac{{AX}}{{AB}}\frac{{{{R}_{w}}}}{{{{R}_{w}}+{{R}_{1}}}}{{E}_{1}}\end{aligned}$