Consider two spherical balls given charges of equal magnitude and opposite signs. Now, we connect them with a conducting straight finite wire. A current will flow in the wire till the charges on the spheres become zero. To find the magnetic field at a point lying on the perpendicular bisector of the wire, we may use Ampere's Law along with Maxwell's correction, since the electric field in changing with time in space.
How should I use Biot Savart Law to obtain the same expression for field? Should a displacement current be considered? Doesn't Biot Savart Law only give the field due to current flowing in conducting wires?
Answer
According to Griffiths and Heald, one can obtain a Biot-Savart-like formula $${\bf B} = \frac{\mu_0}{4 \pi} \int \frac{( {\bf J}+\epsilon_o \frac{\partial {\bf E}}{\partial t})\times \hat{{\bf r}}}{r^2} d \tau,$$ where the integrand is evaluated at the simultaneous (non-retarded) time. However, they point out that this is not useful for practical computation since it is self referential, namely to calcualte ${\bf B}$ everywhere you need to know ${\bf E}$ everywhere, and to know ${\bf E}$ everywhere you need to know ${\bf B}$ everywhere (because of Faraday induction, ${\bf \nabla} \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}$).
Heras commented that the more useful generalisation to the Biot Savart Law for time dependent sources was given by Jefimenko (Electricity and Magnetism, 2nd Ed., Electrect Scientific, Star City, 1989 p 516) and by Jackson (Classical Electrodynamics, 2nd Ed., Wiley, New York, 1975, p 225) and is given as $${\bf B} = \frac{\mu_0}{4 \pi} \int d^3x' \left( \frac{[{\bf J} \times {\hat {\bf R}}]}{R^2} + \frac{[\partial {\bf J}/\partial t] \times {\hat {\bf R}}}{Rc} \right),$$ where the square brackets indicate that the enclosed quantity is to be evaluated at the retarded time $t'=t−R/c$.
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