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A scalar potential of a field is equal to 1/r. Can it be magnetic field? Why?

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Question

A scalar potential of a field is equal to 1/r. Can it be magnetic field? Why?

Expert Solution

Step 1

The divergence of magnetic field whether it be due to steady or unsteady current source is always zero,

$\nabla \cdot \vec{B} = 0 (1)$

As the divergence of a curl of any vector field is also zero, it follows then, equation (1) can always be written as,

$\nabla \cdot \vec{B} = \nabla \cdot (\nabla \times \vec{A}) = 0 \vec{B} = (\nabla \times \vec{A}) (2)$

The vector $\vec{A}$ is called the vector potential, the curl of which gives the magnetic field, this always holds for magnetic fields.

An analogous quantity exist for electric fields called scalar potential V, gradient of V gives the electric field, mathematically,

$\vec{E} = - \nabla V (3)$

This is motivated by the fact that curl of electric field due to stationary charges is zero, and hence can be written as gradient of a scalar function(since curl of gradient is always zero) i.e.

$\nabla \times \vec{E} = 0 (4)$

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