Analog to Biot-Savart for E field

• May 26, 2018

• #1

scoomer

18

4

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.

 

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• May 26, 2018

• #2

Nguyen Son

53

35

Hope it can help you

 

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• May 26, 2018

• #3

Gene Naden

321

64

##\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.

 

• May 26, 2018

• #4

jtbell

Staff Emeritus

Science Advisor

Homework Helper

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scoomer said:

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.

 

• May 27, 2018

• #5

scoomer

18

4

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

 

• Jun 17, 2023

• #6

Just Someone Online

1

0

Gene Naden said:

##\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

 

1. What is the Analog to Biot-Savart law for electric fields?

The Analog to Biot-Savart law for electric fields is a mathematical equation that describes the relationship between the electric field at a point in space and the current that produces it. It is similar to the Biot-Savart law for magnetic fields, but applies to electric fields instead.

2. How is the Analog to Biot-Savart law derived?

The Analog to Biot-Savart law is derived from the principles of electrostatics and Gauss's law. It involves integrating the electric field contribution from each small segment of a current-carrying wire to determine the total electric field at a point.

3. What are the assumptions made in the Analog to Biot-Savart law?

The Analog to Biot-Savart law assumes that the current is steady, the wire is infinitely thin, and the distance between the wire and the point where the electric field is being measured is much greater than the length of the wire. It also assumes that the wire is straight and that the electric field is being measured at a point far away from the wire.

4. Can the Analog to Biot-Savart law be used to calculate the electric field for any current-carrying wire?

No, the Analog to Biot-Savart law can only be used to calculate the electric field for simple geometries, such as straight wires and circular loops. For more complex geometries, other methods such as Coulomb's law and superposition must be used.

5. How is the Analog to Biot-Savart law used in practical applications?

The Analog to Biot-Savart law is used in many practical applications, such as calculating the magnetic fields produced by power lines and electronic devices, designing electromagnets and motors, and studying the behavior of charged particles in electric and magnetic fields.