Line of charge as a volume charge dist. (w/ Dirac delta fcn.)

[AxiomOfChoice]

How would you write an infinite line charge with constant charge per unit length $\lambda$ as a volume charge density using Dirac delta functions? Perhaps in cylindrical coordinates?

I'm confused because if you integrate this charge distribution over all space, you should get an infinite amount of charge...right?

And is there an easy way to do the same thing for a line of charge of length $L$ and $\lambda = Q/L$?

 

[Born2bwire]

You could define it as a piecewise function that uses Dirac deltas. You could define it as the superposition of two appropriate step functions that scale a Dirac delta.

*shrug*

 

[astrokreng]

I can provide a response to the content mentioned above. The concept of a line charge as a volume charge distribution with Dirac delta functions is a mathematical representation used to describe the distribution of charge along an infinitely long line. This representation is commonly used in electromagnetism and allows us to calculate the electric field and potential associated with the line charge.

To write an infinite line charge with a constant charge per unit length \lambda as a volume charge density using Dirac delta functions, we can use the following expression in cylindrical coordinates:

\rho(\rho,z) = \lambda \delta(\rho) \delta(z)

Here, \rho represents the radial distance from the line charge and z represents the distance along the line charge. The Dirac delta function, \delta(x), is a mathematical function that is zero everywhere except at x=0, where it is infinite. This allows us to represent an infinitely thin line of charge with a finite value of \lambda.

When we integrate this charge distribution over all space, we do get an infinite amount of charge. However, this is expected as we are dealing with an infinitely long line charge. This representation is used in theoretical calculations and can be approximated in real-world scenarios by considering a finite length of the line charge.

For a line of charge with a finite length L and a constant charge per unit length \lambda = Q/L, we can use the following expression:

\rho(\rho,z) = \frac{Q}{L} \delta(\rho) \delta(z) \Theta(L-z)

Here, \Theta(x) is the Heaviside step function which is zero for x<0 and 1 for x>0. This function allows us to restrict the charge distribution to a finite length L.

In summary, the concept of representing a line charge as a volume charge distribution using Dirac delta functions is a useful tool in theoretical calculations. It allows us to mathematically describe the distribution of charge along an infinitely long line and can be approximated for finite line charges by incorporating the Heaviside step function.