Electric and magnetic fields in various static situations

Case 1: Electric field due to spherically symmetric charge distribution

  1. Calculating the field via the potentials (calculate $\phi$ then $\vec E=-\nabla\phi$)

  2. Calculating the field via Gauss’ law.

Magnetic field due to current carrying wire

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Magnetic field generated by a wire in $\hat z$ carrying current $I$. The wire is infinitely long

  1. Using Biot-Savart law

  2. We try with Ampere’s law

Charged circular loop

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Consider a charge circular loop of radius $R$ and char per unit length $\lambda$ on the $x$-$y$ plane and centered at the origin. We cant use Gauss’ law because there is no symmetry.

$$ \vec E(x,y,z)=\frac{\lambda Rz}{2\epsilon_0 (R^2+z^2)^\frac 32}\begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} $$

Electric & magnetic dipoles

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🌴 Consider the electrostatic potential and electric field at some point $\vec r$ which is a long way away from some localized distribution of charge.