Resistance

Ohms Law

For many materials, the relationship between current and voltage is linear.

V = IR

For a given voltage, less current flows as the resistance rises.

Wire resistance

We can calculate the resistance of a conductive object if we know its dimensions and properties.

R = resistivity \cdot \frac{length}{area} R = \rho \frac{l}{A}

  • The resistance of a wire is proportional to
    • the resistivity of the material
    • the length of the wire
  • It is inversely proportional to
    • the cross-sectional area

Wire resistance

Electrical energy converted to heat in a wire is usually considered wasted energy.

This energy loss is seen as a reduced voltage across the load or device.

  • Resistivity - property of the material - intensive
  • Resistance - property of the wire - extensive

Units

  • To get proper units of resistance in ohms
  • Resistivity is expressed in Ohm/meter
  • Length in meters
  • Area in square meters

Energy Dissipation in a Resistor

As current flows through a resistor, the potential energy of the charges is converted to heat.

The power converted to heat in the resistor is the voltage across the resistor multiplied by the current through the resistor.

P = IV

The voltage and current through the resistor is related by Ohm’s Law V=IR. So, I can substitute the voltage in the equation and get

P = I (IR) P = I^2R

The power is proportional to the square of the current. This squared relationship is important for many circuit designs. When the current is in amps and the resistance is in ohms, the power is in watts.

We can do the same thing but use Ohm’s Law to substitute for the current and we get

P = \frac{V^2}{R}

Note that the power is not linear with the current or voltage but has a squared relationship.

Similarities to Thermal Conduction

The electrical resistance of a block of material is given by

\textrm{resistance} = \frac{\textrm{resistivity of material}\cdot\textrm{length of material}}{\textrm{cross sectional area of material}}

R = \frac{\rho \cdot l}{A}

The UA of a block of material has the formula

\textrm{UA value} = \frac{\textrm{thermal conductivity of material} \cdot \textrm{cross sectional area}}{\textrm{thickness of material}}

UA = \frac{k \cdot A}{t}

These are basically reciprocals of each other.