13.7.2 Potential Divider Principle

Two resistors R1 and R2 are connected in series. If the total voltage across them is Vtotal, 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}

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