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Derivation Electric Power

Simple circuit with one resistor
Level 2 (suitable for students)
Level 2 requires school mathematics. Suitable for pupils.
Updated by Alexander Fufaev on

When charge \( Q \), due to an applied voltage \( U \) is transported in the conductor, potential energy is converted into kinetic energy. The kinetic energy \( W \) that a charge gains (\(W\) positive)) or loses (\(W\) negative)) by passing through the voltage \(U\) is given by:

Electrical energy
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The power \(P\) is defined as the converted energy \(W\) per time period \(t\):

Definition of power quantity
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The electric power is obtained by substituting Eq. 1 into 2:

Electric power using charge, voltage and time
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The electric current \(I\), is the charge \(Q\) transported per time interval \(t\): \(I = Q/t \). The factor \(Q/t\) is in Eq. 3, so we replace it with \(I\) to eliminate the unknown and experimentally not easily accessible time \(t\). Thus, the electric power becomes:

Electric power (with U, I)
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For an Ohmic conductor (these are those conductors for which Ohm's law applies), Equation 4 can be rewritten using \( U = R \, I \). The power \(P\) can therefore be expressed by the resistance \(R\) of the conductor (or a load) and the applied voltage \(U\):

Electric power (with U, R)
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Illustration : A simple circuit.
Example: Constant voltage

If the voltage \(U\) is kept constant, the regions of the conductor that have the smallest resistance \(R\) will be the warmest, because that is where the converted power is the largest.

Of course you can also rearrange Ohm's law \( U = R\,I \) with respect to the current \( I = U/R \), and use it to rewrite the electric power:

Electric power (with I, R)
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Example: Constant current

When the current \(I\) is kept constant, the regions of the conductor that have the largest resistance \(R\) will be the warmest, because that is where the power converted is the largest.

Illustration : The resistance \(R_2\) is larger than \(R_1\) and heats up more at a constant current \(I\).