Dirac Delta function and charge density.

[calcisforlovers]

I have a line charge of length L and charge density /lambda on the Z-axis. I need to express the charge density in terms of the Dirac Delta function of theta and phi. How would I go about doing this?

 

[pam]

$$\rho=\lambda\delta(1-\cos\theta)U(L-r))/(2\pi r^2)$$,
where U is the unit step function, should be the charge density in spherical coordinates.

 

[astrokreng]

The Dirac Delta function is a mathematical tool used to represent a point-like charge or distribution in space. In this case, it can be used to express the charge density of a line charge on the Z-axis. To do this, we first need to define the charge density as a function of the position vector (r) in spherical coordinates (r, theta, phi). We can then use the Dirac Delta function to represent the charge density as a point charge at the origin (r=0) in the theta and phi directions.

The charge density (/lambda) can be expressed as:

/lambda(r, theta, phi) = /lambda_0 * delta(theta) * delta(phi)

where /lambda_0 is the charge density at the origin and delta(theta) and delta(phi) are the Dirac Delta functions in the theta and phi directions, respectively.

To express the charge density in terms of the Dirac Delta function, you would need to integrate the charge density over the length of the line charge (L) in the z-direction. This would give you:

/lambda_0 = integral from -L/2 to L/2 of /lambda(r, theta, phi) dz

Substituting this into the expression above, we get:

/lambda(r, theta, phi) = integral from -L/2 to L/2 of /lambda(r, theta, phi) dz * delta(theta) * delta(phi)

This expression allows us to represent the charge density in terms of the Dirac Delta function of theta and phi. It is important to note that the Dirac Delta function is only defined for real values of its argument, so the integration limits should be chosen accordingly.

In summary, the Dirac Delta function can be used to represent a point-like charge or distribution in space, and in this case, it can be used to express the charge density of a line charge on the Z-axis. By integrating the charge density over the length of the line charge, we can express the charge density in terms of the Dirac Delta function of theta and phi.