# Solving Griffiths' Electrodynamics Ex. 10.2

• LuxAurum
In summary, at point P away from the wire, the electric and magnetic fields are zero because the wave has not had enough time to reach the point. As the time increases, the fields gradually become positive until the wave reaches the point.
LuxAurum

## Homework Statement

I'm working through the 3rd edition of Griffiths' Electrodynamics book and have gotten stuck on some details in example 10.2, which describes an infinite straight wire carrying a current I0 for t>0.

The figure included with the example illustrates a wire in the vertical z direction with element dz. A point P is located at distance s from the wire. The distance from dz to P is given by script_r, and forms a right triangle with base of z.

The part that is unclear to me is the following. Griffiths states:
"For t < s/c, the 'news' has not yet reached P and the potential is zero. For t > s/c, only the segment |z| $\leq \sqrt{(ct)2 - s2}$
contributes (outside this range tr is negative..."

## Homework Equations

tr = t - (script_r/c)

## The Attempt at a Solution

I understand that the expression t = s/c represents the time it takes for the wave to traverse the distance s to get to P. What I'm unclear about is how to properly setup and analyze this problem if it wasn't an example. Is the distance s used to establish a lower bound for the time? Griffiths then uses (ct)2 in the expression for |z|.

Also, why does I(tr) = 0 when tr is negative?

Hopefully this doesn't come across as scattered. Sometimes I get stuck on minor details that should be obvious but aren't at the time.

In this problem we are to derive the electric and magnetic fields at any point P away from an infinite straight wire carrying a constant current I0.
Generally, E = -∇V - ∂A/∂t and B = ∇ x A
Since the wire has equal positive and negative charges, V = 0 everywhere outside the wire. Once we determine A then the above eqn.s allow us to determine E and B.
I assume you agree with their eqn. for A(s,t) at point P, which was derived from eqn. 10.19 referring to Fig. 10.4 analysis.
As you noted the retarded time, tr = t - (script_r/c) ,where (script_r/c) is the time delay for the source (in this case the dz contribution of the current I0) to reach P traveling at speed c. The first potential contribution reaches P when t = s/c. Before that time A = 0. As t increases, the potential contributions are from +z to –z only, because beyond those the contributions haven’t had enough time to arrive (that is, their tr is negative, and hence their contribution to A is zero).
The integration for A(s,t) at P from + z to – z is determined as shown, and E and B are thus determined as a function of t. From the results you can see how E = 0 and B is as noted as t approached infinity.

Thank from my side as well

## 1. How do I approach solving Griffiths' Electrodynamics Ex. 10.2?

The best approach to solving this problem is to first carefully read the question and understand what is being asked. Then, review relevant concepts and equations from the textbook or lecture notes. Finally, break the problem down into smaller, manageable steps and use logical reasoning to arrive at a solution.

## 2. What are the key equations I need to know for solving this problem?

The key equations for this problem are Gauss's Law, Ampere's Law, and the Biot-Savart Law. It is also important to understand the concept of electric flux and how it relates to electric field and charge distribution.

## 3. How do I know which mathematical techniques to use for solving this problem?

The mathematical techniques you should use will depend on the specific details of the problem. In general, it is helpful to first draw a diagram and label all given quantities. Then, use your understanding of the relevant equations and concepts to determine which mathematical techniques are most appropriate.

## 4. What are some common mistakes to avoid when solving this problem?

Common mistakes when solving this problem include forgetting to properly account for the direction of electric field and charge distribution, using incorrect units, and not checking the final answer for reasonableness. It is also important to double check all calculations and make sure all steps are clearly shown.

## 5. How can I check my answer to make sure it is correct?

The best way to check your answer is to go back through your steps and make sure they are all correct. Look for any potential errors or mistakes, and double check your calculation methods. You can also compare your answer to the solution provided in the textbook or by your instructor. If possible, it is also helpful to ask a classmate or instructor to review your work and provide feedback.

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