Ohm's Law

Double Resistance, Half Current

current reverses resistance

With voltage held fixed, current is inversely proportional to resistance. Double the resistor and the current halves, here to an exact fraction.

electricity Same 12 V battery, resistance doubled from 4 ohm to 8 ohm: the current halves from 3 A to 3/2 A, kept exact as a fraction with no decimals.

Example

With voltage held fixed, current is inversely proportional to resistance. Double the resistor and the current halves, here to an exact fraction. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Recall the flagship circuit

Start from the Ohm's law flagship: 12 volts across 4 ohm gives a current of 3 amperes.

I=12 V4 ohm=3 AI = \frac{12\ \text{V}}{4\ \text{ohm}} = \hl{3}\ \text{A}
The starting circuitA loop with a battery, a resistor, and the ammeter reading the current.12 VA3 A4 ohm3 A

Now double the resistance

Keep the same 12 volt battery, but double the resistor to 8 ohm. The ammeter reading is what we want to find.

R→8 ohm,V unchangedR \rightarrow 8\ \text{ohm}, \quad V \text{ unchanged}
The same battery, a larger resistorThe same loop with the battery unchanged and a larger resistor; the ammeter reading is to be found.12 VA8 ohm

Apply Ohm's law to the new resistance

Divide the unchanged voltage by the new, larger resistance.

I=12 V8 ohmI = \frac{12\ \text{V}}{8\ \text{ohm}}

Fixed voltage makes current fall as resistance rises

Now compare three resistors while the battery stays at 12 volts. The larger resistor gives the same push less room to move charge, so the current falls.

VRI12 V3 ohm4 A12 V4 ohm3 A12 V8 ohm32 A\begin{array}{c|c|c}V & R & I \\12\ \text{V} & 3\ \text{ohm} & 4\ \text{A} \\12\ \text{V} & 4\ \text{ohm} & 3\ \text{A} \\12\ \text{V} & 8\ \text{ohm} & \tfrac{3}{2}\ \text{A}\end{array}
Same battery, compared resistorsThe same loop with the battery held fixed while the table compares different resistor values.12 VA3/2 A8 ohm3/2 A

Reduce the fraction exactly

The fraction 12 over 8 reduces by dividing top and bottom by 4, leaving the exact current with no decimals.

128=32\frac{12}{8} = \frac{3}{2}

The current halves

The current is 3 over 2 amperes, exactly half of the 3 amperes from before. With voltage held fixed, current is inversely proportional to resistance: double the brake, halve the flow. The fraction stays exact.

I=32 A=12×3 AI = \hlmath{\tfrac{3}{2}}\ \text{A} = \tfrac{1}{2} \times \hl{3}\ \text{A}
Twice the resistance, half the currentThe loop with the larger resistor; the ammeter and arrow read the new, smaller current as an exact fraction.12 VA3/2 A8 ohm3/2 A