Find the density of surface charges at the boundary of two conductors with different resistivities

[tellmesomething]

Homework Statement

Two cylindrical conductors with equal cross sections and different resistivities p1 and p2 are put end to, end as shown in figure-3.11.
Find the density of surface charges at the boundary of the conductors if a current I flows from conductor I to conductor 2 .

Relevant Equations

None

I tried using gauss law. This is how i set it up: I took a gaussian pill box of same area and very small length dx, the diagram is amplified for obvious reasons.

Assuming there's a constant field in the yellow portion $E_{1}$ and a constant field $E_{2}$ in the green portion. I assume the area of cross section as $A$ and surface charge density as $\sigma$

$$(\frac{\rho_{2} I}{A} - \frac{\rho_{1} I}{A})A = \frac{\sigma A}{\epsilon_{0}}$$

Is this valid? Because it doesnt get me the answer as i do not know what $A$ is.

i

 

[TSny]

Your work looks correct

$$(\frac{\rho_{2} I}{A} - \frac{\rho_{1} I}{A})A = \frac{\sigma A}{\epsilon_{0}}$$

This looks good.

Is this valid? Because it doesnt get me the answer as i do not know what $A$ is.

The answer for $\sigma$ will depend on the cross-sectional area $A$. The problem statement is a little inconsistent in not providing a symbol for the cross-sectional area while it does provide symbols for the current and resistivities.

 

[kuruman]

You have derived the boundary condition for the discontinuity of the normal component of the electric field at the interface between conductors. That is correct. You are also correct in saying that you need the area. Think of it this way, two setups like this carrying the same current $I$ but having different cross sectional areas will have different surface charge densities at the interface.

On edit:
I see that @TSny has preempted me (again.) I will leave this post up as additional confirmation.

 

[tellmesomething]

Your work looks correct

This looks good.


The answer for $\sigma$ will depend on the cross-sectional area $A$. The problem statement is a little inconsistent in not providing a symbol for the cross-sectional area while it does provide symbols for the current and resistivities.

Yes i did a unit check as well and the answer given in the key is $\epsilon_0(\rho_{2}-\rho_{1})I$ , the unit comes out to be Coulumb instead of Coulumb per sqaured meter. I guess they meant charge instead of charge density.

 

[tellmesomething]

You have derived the boundary condition for the discontinuity of the normal component of the electric field at the interface between conductors. That is correct. You are also correct in saying that you need the area. Think of it this way, two setups like this carrying the same current $I$ but having different cross sectional areas will have different surface charge densities at the interface.

On edit:
I see that @TSny has preempted me (again.) I will leave this post up as additional confirmation.

Yes thankyou :)

 

[haruspex]

I guess they meant charge instead of charge density.

Or current density instead of current?

 

[tellmesomething]

Or current density instead of current?

Yes possible