How Can Lambda be Derived from Gauss's Law for an Infinite Line of Charge?

[beer]

Given the following information:
An infinite line of charge lies long the z-axis. The electric field a perpendicular distance 0.490m from the charge is 770N/C .

We're asked to do the following:
How much charge is contained in a section of the line of length 1.20cm ?

The answer is 2.52*10^-10 C. I reached the answer by plugging the known values into the following formula:
q = 2π r L ε₀ E

That formula was given on a website. And it kind of makes sense.

I'd like to be able to derive THAT formula from the following formula:
E = (1 / 2 π ε₀) (λ / r)

Which is the general formula for finding the electric field in both a cylinder or a wire. I'm getting tripped up on lambda. Is it possible to derive the second formula from the first? I'm having a hard time maniuplating lambda to make it work. I know lambda is charge per unit length, but can someone help resolve this for me?

 

[beer]

This was moved but it isn't a homework question. I answered the question already...

I'm looking for discussion on ways to deduce one formula from the other. This has little to do with the question. Can this be moved back to the correct section?

 

[gneill]

This was moved but it isn't a homework question. I answered the question already...

I'm looking for discussion on ways to deduce one formula from the other. This has little to do with the question. Can this be moved back to the correct section?

It is a homework-type question; You're asking how to derive a formula. The homework section is appropriate.

Did you try solving your second equation for λ? Since λ is the charge per unit length, if you multiply λ by a given length you get the amount of charge associated with that length.

 

[beer]

The only time I've repeatedly used lambda was in the study of mechanics - primarily statics and the like, where lambda was, essentially, a vector divided by it's magnitude.

In the case of my post above, what is lambda equal too?

λ = E 2π ε₀r

That's just algebraic rearrangement, though. In the first formula in the original post, lambda isn't included in the equation. Instead, there is L, representing length. My question is - are the two formulas related? If so, there is some connection between them, lambda, and length. It seems like an obvious connection, but I can't deduce one equation from the other.

I ask because the question is conceptually simple, but I'd not have been able to solve it with the equations I initially had at hand. (I'm probably over looking something rather obvious.)

 

[beer]

Essentially, it would seem, if the two are connected then lambda is equal to some arrangement of q and length. Charge per unit length. q/L perhaps.

Is it really that simple?

 

[molsen]

Yes :) I cannot see why you doubt it.

E = (1 / 2 π ε0) (λ / r)

Rewrite by multiplying the two fractions:

E = (1 λ) / (2 π ε0 r)

Multiply with the denominator on both sides:

2 π ε0 r E = λ

Rearrange:

2π r ε0 E = λ

Compare to the other formula:

q = 2π r L ε0 E

In this, divide by L on both sides:

q / L = 2π r ε0 E

Insert the first formula:

q / L = 2π r ε0 E = 2π r ε0 E = λ

So: q / L = λ

In other words, λ is charge (q) per length (L) as qneill mentioned.

 

[jtbell]

Is it really that simple?

Sometimes things in physics really do turn out to be simple. :D

If charge is uniformly distributed along a line, then λ = q/L has the same value for any section of any length L, containing a corresponding amount of charge q.

 

[beer]

It's always a pleasure to check old replies and see what I thought was confusing a few weeks ago now makes great sense. :)

Thanks for the help guys! I have my electrostatics exam tomorrow night. I feel mostly prepared for it.