I Analog to Biot-Savart for E field

[scoomer]

On page 317 of "INTRODUCTION TO ELECTRODYNAMICS" 4th Ed. by Griffiths he states without proof that the analog to Biot-Savart for finding the E field is:

Can anyone direct me to a reference where this is proved or give me a hint how to prove it? Thank you.

 

[Nguyen Son]

Hope it can help you

 

[Gene Naden]

$\nabla \cdot E=\frac{1}{\epsilon}\rho$
Gauss's law $E=\frac{1}{4\pi\epsilon}\int \frac{(\nabla \cdot E) \hat r}{r^2} d \tau$
Use the identity $A \times (B \times C) = B (A \cdot C)-C(A\cdot B)$ where $A=\nabla$, and possibly integrate by parts, I am not sure,
You get an an integral like $\frac{1}{4\pi}\int \frac{(\nabla \times E) \times \hat r}{r^2} d\tau$.
Use Maxwell's equation $\nabla \times E = -\frac {\partial B}{\partial t}$.
This gives you $E=-\frac {1}{4\pi} \int \frac {(\frac {\partial B}{\partial t}) \times \hat r}{r^2} d\tau$
Since you are integrating over space and not time, we can move the partial derivative out of the integral, getting
$E=-\frac {1}{4\pi} \frac{\partial}{\partial t} \int \frac { B \times \hat r}{r^2} d\tau$
I hope this helps.

 

[jtbell]

On page 317 of "INTRODUCTION TO ELECTRODYNAMICS" 4th Ed. by Griffiths

I don't have the 4th edition. In the 3rd edition, your equation appears as part of Problem 7.47:

Use the analogy between Faraday's Law and Ampère's law, together with the Biot-Savart law, to show that <<your equation>> for Faraday-induced electric fields.

So if you assume the validity of Biot-Savart, you can start from there and use the similarity between Faraday and Ampère to suss out the analogous equation for $\vec E$. Or you can look for a derivation of Biot-Savart from Maxwell's equations, and then adapt that to work for $\vec E$ instead of $\vec B$, again using the similarities between the various Maxwell equations.

 

[scoomer]

Thanks to Nguyen Son, Gene Naden and jtbell. Your responses are very helpful and much appreciated.

 

[Just Someone Online]

$\nabla \cdot E=\frac{1}{\epsilon}\rho$
Gauss's law $E=\frac{1}{4\pi\epsilon}\int \frac{(\nabla \cdot E) \hat r}{r^2} d \tau$
Use the identity $A \times (B \times C) = B (A \cdot C)-C(A\cdot B)$ where $A=\nabla$, and possibly integrate by parts, I am not sure,
You get an an integral like $\frac{1}{4\pi}\int \frac{(\nabla \times E) \times \hat r}{r^2} d\tau$.
Use Maxwell's equation $\nabla \times E = -\frac {\partial B}{\partial t}$.
This gives you $E=-\frac {1}{4\pi} \int \frac {(\frac {\partial B}{\partial t}) \times \hat r}{r^2} d\tau$
Since you are integrating over space and not time, we can move the partial derivative out of the integral, getting
$E=-\frac {1}{4\pi} \frac{\partial}{\partial t} \int \frac { B \times \hat r}{r^2} d\tau$
I hope this helps.

I think so $E=\frac{1}{4\pi\epsilon}\int \frac{(\nabla \cdot E) \hat r}{r^2} d \tau$

Reference: https://www.physicsforums.com/threads/analog-to-biot-savart-for-e-field.948269/ equation will give you 0 as induced electric fields have 0 curl so this eq is not valid here