# Does the potential difference across a resistor depend on current?

As context, please consider a battery with electromotive force $$\mathcal{E}$$ connected to a resistor $$R$$ and a switch $$S$$ which connects and disconnects the resistor from the opposite terminal of the battery. If the switch is closed, by Kirchhoff's loop rule the resistor causes a drop in voltage equal to the potential difference of the battery. However, if the switch is open the voltage difference seemingly disappears across the resistor, and the potential difference across the switch is now equivalent to $$\mathcal{E}$$. Essentially, does the voltage drop across a resistor depend if current is passing through it?

Yes, this is exactly what Ohm's Law says:

$$V=IR$$

for a potential difference $$V$$, current $$I$$ and resistance $$R$$.

• I would think it is the other way around: potential differences cause current. Or is there really no distinction? – BioPhysicist May 3 '20 at 20:57
• @AaronStevens Whether one is the "cause" of the other depends on what you're manipulating in the circuit. If you manipulate the voltage of a constant-voltage power supply, then increasing the voltage causes the current across the resistor to increase. If you manipulate the current of a constant-current power supply, then increasing the current causes the potential difference across the resistor to increase. – probably_someone May 3 '20 at 21:06
• @AaronStevens Meanwhile, if you manipulate the resistance of the circuit, then what happens depends on what kind of power supply you have. For a constant-voltage power supply, increasing the resistance causes the current through the resistor to decrease. For a constant-current power supply, increasing the resistance causes the potential difference across the resistor to increase. – probably_someone May 3 '20 at 21:10
• @aaronStevens, To add: in dynamic systems modeling, a purely resistive component can exhibit either differential or integral causality depending on whether it is being driven by an effort source or a flow source. This is not true of inertances or compliances i.e., you cannot instantaneously assert an effort upon a compliance (that would require infinite flow) nor can you instantaneously assert a flow upon an inertance (which would require infinite effort). – niels nielsen May 4 '20 at 2:16
• @AaronStevens Don't think at Ohm's law or at any other element constitutive relationship as cause-effect. Constitutive laws, like the Ohm's law, just express constraints between the element's terminal variables, specifying a set of admissible voltages and currents. – Massimo Ortolano May 4 '20 at 6:08

According to Ohm's Law, electric current through a conductor between two points is directly proportional to the voltage across the two points i.e. $$I\propto V$$ Or$$V\propto I$$ It is clear from above relation that the voltage $$V$$ across a resistor (conductor) is directly proportional to the electric current $$I$$ passing through it i.e. higher the electric current $$I$$, greater is the potential difference $$V$$ keeping resistance constant