A charge of 0.5 μC is located at A (10,-10,10) (in cm) and a second charge of -0.3 μC is located at B (-10,15,5) (in cm). Find E (Electric Field) at P (15,20,50) (in cm).

The electric field at E at point P is represented by .

Explanation:

The definition of the electric field is represented by the following differential equation:

Where:

- Electric field, measured in newtons per coulomb.

- Electrostatic force, measured in newtons.

- Theoretical electric charge, measured in coulombs.

Electrostatic force is modelled by Coulomb's Law, which states that such force is directly proportional to electric current and inversely proportional to square distance. In this case, vectorial form of this law is:

Where:

- Electrostatic constant, measured in newton-square meters per square coulomb.

- Real electric charge, measured in coulombs.

- Magnitude of the vector position, measured in meters.

- Vector position, measured in meters.

We get the electric field equation by deriving the equation above:

The total electric field is obtained by means of this concept and the Principle of Superposition as well:

At first we determine each vector position and their magnitudes:

If we know that , , , , , and , then we get the electric field at point P.

The electric field at E at point P is represented by .

i think it because equal matrices have equal dimensions, only square matrices can be symmetric. for all indices and . every square diagonal matrix is symmetric, since all off-diagonal elements are zero. in linear algebra, a real symmetric matrix represents a self-adjoint operator over a real inner product space.

where are the vectors ?