Conductor resistance and rheostat

Two wires of the same metal may carry different currents at the same voltage. What happens if a wire is made longer or thicker, and how does a rheostat's slider control this?

Imagine adding an identical second piece to a wire, extending its single current path. Electrons interact with the material along a longer path, so more voltage is needed to maintain the same current. At unchanged material, cross-sectional area and temperature, resistance is proportional to length l: doubling length doubles R; halving length halves R. Length is measured in metres. We will derive the formal connection rules next lesson.

Keep length unchanged but increase the cross-sectional area. Imagine more identical narrow paths side by side: at the same voltage, more charge passes through the whole wire each second. Resistance is therefore smaller. Area S is measured in m² and refers to a cut across the wire, not its side surface. Doubling S halves R; doubling a round wire's diameter makes its area four times, not twice, as large.

Compare a copper wire with a wire of another material, with equal lengths and cross-sections. Their resistance difference now comes from the materials. Resistivity ρ characterises this at a given temperature; its SI unit is ohm metre. Greater ρ means greater R for otherwise identical wires. Here ρ is an electrical property, not mass density. Units help distinguish them: ohm metres cannot be confused with kg/m³.

The relationships combine as R = ρl/S: length and resistivity increase the result; area decreases it. This applies to a uniform wire of constant cross-section at a uniform, unchanged temperature. The units give Ом·м · м/м² = Ом. Shortening the wire reduces its own resistance contribution. As S approaches zero, the model predicts a large R, but particle sizes and heating limit this extrapolation in reality.

Wire areas are often conveniently measured in mm². Tables then give ρ in ohm·mm²/m while length stays in metres; these units consistently cancel to ohms. For SI, convert 1 мм² = 10⁻⁶ м², because the factor 10⁻³ must also be squared. Assuming that area, like length, changes by one thousandth gives an answer wrong by a factor of a thousand. Choose one unit set before substituting.

A rheostat does not stretch its wire or change the material. Its slider selects the length of winding connected between the circuit terminals. For adjustment, connect one winding end and the slider terminal: current passes through the section between them. If only the two ends are connected, the full length is always used, wherever the slider moves. Trace the actual current path before predicting the resistance change.

Moving the slider away from the connected end increases the active length and thus the rheostat's resistance. With unchanged source voltage, current decreases in a simple circuit with a single path. Moving it nearer reverses the effect. Before switching on a classroom circuit, select a high permitted resistance and check the current rating: excessive current can heat the winding. Slider direction alone tells us nothing unless the connected end is specified.

ρ = 0,5 Ом·мм²/м, l = 10 м, S = 2 мм². Find R.

Use the uniform-wire formula R = ρl/S. The units already match: ohm·mm²/m is multiplied by m and divided by mm².

R = 0,5·10/2 = 2,5 Ом. No conversion to m² is needed here; it would require converting resistivity at the same time.

Twice the length at unchanged cross-section, material and temperature would give 5 ohms. This agrees with a longer path having greater resistance.

2.5 ohms: the resistance of this wire at the temperature for which ρ is given.

Geometry explains R = ρl/S: longer means greater R; more cross-sectional area means smaller R.

The units of ρ and S must belong to the same consistent set.

A rheostat's slider changes the connected length; the terminal connections determine the direction of the effect.

A rheostat and a load are often in series. We will derive the rules for that connection.