Superposition Principle: Difference between revisions
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===A Mathematical Model=== | ===A Mathematical Model=== | ||
The Superposition Principle is derived from the properities of additivity and homogeneity for linear systems which are defined by | |||
<math>F(x_1 + x_2) = F(x_1) + F(x_2)</math> | |||
<math>aF(x) = F(ax)</math> | |||
for a scalar value of a. The principle can be applied to any linear system and can be used to find the net result of functions, vectors, vector fields, etc. For the topic of introductory physics, it will mainly apply to vectors and vector fields such as electric forces and fields. | |||
If given a number of vectors passing through a certain point, the resultant vector is given by simply adding all the the vectors at that point. For example, for a number of uniform electric fields passing though a single point, the resulting electric field at that point is given by | If given a number of vectors passing through a certain point, the resultant vector is given by simply adding all the the vectors at that point. For example, for a number of uniform electric fields passing though a single point, the resulting electric field at that point is given by | ||
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<math>\vec{E} = \frac{1}{4 \pi \epsilon_0}\sum_{i=1}^n\frac{q_i}{r_i^2}\hat{r_i}</math> | <math>\vec{E} = \frac{1}{4 \pi \epsilon_0}\sum_{i=1}^n\frac{q_i}{r_i^2}\hat{r_i}</math> | ||
This approach can be applied to any other sources of electric or magnetic field or force by simply adding together the each of the vectors at a specific point | |||
==Examples== | ==Examples== | ||
Revision as of 00:59, 5 December 2015
The Superposition Principle states that the net result of multiple vectors acting on a given place and time is equal to the vector sum of each individual vector. For intro physics, this mostly relates to effect that multiple electric or magnetic fields and forces have on a certain location.
A Mathematical Model
The Superposition Principle is derived from the properities of additivity and homogeneity for linear systems which are defined by
[math]\displaystyle{ F(x_1 + x_2) = F(x_1) + F(x_2) }[/math] [math]\displaystyle{ aF(x) = F(ax) }[/math] for a scalar value of a. The principle can be applied to any linear system and can be used to find the net result of functions, vectors, vector fields, etc. For the topic of introductory physics, it will mainly apply to vectors and vector fields such as electric forces and fields.
If given a number of vectors passing through a certain point, the resultant vector is given by simply adding all the the vectors at that point. For example, for a number of uniform electric fields passing though a single point, the resulting electric field at that point is given by
[math]\displaystyle{ \vec{E} = \vec{E}_{1} + \vec{E}_{2} +...+ \vec{E}_{n} = \sum_{i=1}^n\vec{E}_{i} }[/math]
and this same concept can be applied to electric forces as well as to magnetic fields and forces. This is more useful when dealing with the effect that multiple point charges have on each other is an area of void of other electric fields. The resultant electric field for at a point for a system of point charges is given by
[math]\displaystyle{ \vec{E} = \frac{1}{4 \pi \epsilon_0}\sum_{i=1}^n\frac{q_i}{r_i^2}\hat{r_i} }[/math]
This approach can be applied to any other sources of electric or magnetic field or force by simply adding together the each of the vectors at a specific point
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