General relationship for direction of E field at any given point

[Ravenatic20]

I was going through my textbook, Introduction to Electrodynamics, and I came across this question that puzzled me. The book is really great by the way, I would highly recommend it. No, this isn't a homework question, it just got me thinking.

For a finite line of charge (like a rod, for example), there should be a general relationship for the direction of the electric field no matter where point X is located with respect to the finite line of charge. What do you think this general relationship is?

Lets assume it’s a finite line of positive charge. I think of the electric field (E-field) always pointing outwards. So if you take a point X directly above the finite line of charge, say centered, it’s going to point up. But what relationship can we use to describe this?

I know you can take a bunch of little dq's and add them up, and the direction each one of those points as X can be added up as the direction of the E-field.

Can we incorporate the right-hand-rule with this? No rush to answer I was just curious.

 

[tiny-tim]

Hi Ravenatic20!

… I think of the electric field (E-field) always pointing outwards. So if you take a point X directly above the finite line of charge, say centered, it’s going to point up. But what relationship can we use to describe this?

I know you can take a bunch of little dq's and add them up, and the direction each one of those points as X can be added up as the direction of the E-field.

Can we incorporate the right-hand-rule with this? No rush to answer I was just curious.

I think you're trying to say that the direction of the field (which is what the question asks for) always points perpendicularly towards or away from the line.

This follows from symmetry.

(the right-hand-rule has nothing to do with this … there's no current in the line )

 

[nnnm4]

tiny-tim, what you say is only if the charge is on an infinite line, but Ravenatic was talking about a finite line of charge.

Anyway, this is a solvable problem. Ravenatic's method of integration is correct, just integrate up the charge density using Coulombs law and you get your E filed. I don't think the integration is trivial, but it can be done. BTW, the right-hand rule has no bearing on this, but it would be if the there was a line of current and you wanted to calculate the B-field.

 

[Born2bwire]

It is, however, a prime candidate for far-field approximation. If we were to observe the fields at a distance r>>L, then the field should be fairly approximated as a point charge of
$$Q = \int_{-L/2}^{L/2} \rho(z)dz$$

 

[tiny-tim]

oops!

tiny-tim, what you say is only if the charge is on an infinite line, but Ravenatic was talking about a finite line of charge..

oops!

I somehow read "finite" as "infinite"