13.7.2 Potential Divider Principle

Two resistors R₁ and R₂ are connected in series. If the total voltage across them is V_total, how is this voltage divided between R₁ and R₂?

Since the current following through them is the same,

$\displaystyle \displaystyle \begin{aligned}{{V}_{1}}&=I{{R}_{1}}\\{{V}_{2}}&=I{{R}_{2}}\\{{V}_{{total}}}&=I({{R}_{1}}+{{R}_{2}})\end{aligned}$

From here, it is obvious that the resistor with the larger resistance grabs a bigger share. In fact, the voltage ratios follow the resistance ratios. This is called the potential divider principle.

$\displaystyle \begin{aligned}\frac{{{{V}_{1}}}}{{{{V}_{{total}}}}}&=\frac{{I{{R}_{1}}}}{{I{{R}_{1}}+I{{R}_{2}}}}\text{ }\Rightarrow \text{ }{{V}_{1}}&=\frac{{{{R}_{1}}}}{{{{R}_{1}}+{{R}_{2}}}}{{V}_{{total}}}\\\frac{{{{V}_{2}}}}{{{{V}_{{total}}}}}&=\frac{{I{{R}_{2}}}}{{I{{R}_{1}}+I{{R}_{2}}}}\text{ }\Rightarrow \text{ }{{V}_{2}}&=\frac{{{{R}_{2}}}}{{{{R}_{1}}+{{R}_{2}}}}{{V}_{{total}}}\end{aligned}$