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Mathematics for Physics

87 Superposition

A linearity principle that is used to obtain the net field or force as the vector sum of independent contributions, where a vector sum is an addition that accounts for both magnitude and direction.

Superposition of electric fields. The net electric field of many charges is the vector sum of the fields that each charge would produce alone. This principle is used to build the field of a complicated charge collection from simpler pieces.

The superposition of electric fields is

\[ \mathbf{E}(\mathbf{r}) = \sum_{i=1}^{n}\mathbf{E}_{i}(\mathbf{r}) \]

where

  • \(\mathbf{E}(\mathbf{r})\) is the net electric field at position \(\mathbf{r}\).
  • \(\mathbf{E}_{i}\) is the electric field of source \(i\).
  • \(n\) is the number of sources.

Superposition of magnetic fields. The net magnetic field of many steady currents is the vector sum of the fields of the individual currents. This principle is used to add the Biot-Savart contributions of several wires.

The superposition of magnetic fields is

\[ \mathbf{B}(\mathbf{r}) = \sum_{i=1}^{n}\mathbf{B}_{i}(\mathbf{r}) \]

where

  • \(\mathbf{B}(\mathbf{r})\) is the net magnetic field at position \(\mathbf{r}\).
  • \(\mathbf{B}_{i}\) is the magnetic field of source \(i\).
  • \(n\) is the number of sources.

Superposition of forces. Independent forces on one charge add as vectors. This principle is used to replace a many-force problem by one net force.

The superposition of forces is

\[ \mathbf{F}_{\mathrm{net}} = \sum_{i=1}^{n}\mathbf{F}_{i} \]

where

  • \(\mathbf{F}_{\mathrm{net}}\) is the net force.
  • \(\mathbf{F}_{i}\) is an individual force.

Linearity of Maxwell’s equations. Maxwell’s equations are linear in the fields and sources, so overlapping electromagnetic waves add field by field. Linearity is the property that a sum of solutions is a solution. This principle is used to analyze interference and standing waves.