Calculate the maximum voltage that can be applied

[xxwwyytt]

Homework Statement

A detector of radiation called a Geiger-Muller counter consists of a closed, hollow, conducting cylinder with a fine wire along its axis. Suppose that the internal diameter of the cylinder is 3.00 cm and that the wire along the axis has a diameter of 0.2 mm. If the dielectric strength of the gas between the central wire and the cylinder is 1.00 ✕ 106 V/m, calculate the maximum voltage that can be applied between the wire and the cylinder before breakdown occurs in the gas.

Homework Equations

E=-∫Vds
∫EdA = Qenc/ε0

The Attempt at a Solution

Since the electric field is not constant between the cylinder and the wire, I tries to derive for the electric field using Gauss' law from the wire to the cylinder using radius 1.5e-2m and 0.1e-3m. However the equation comes out like Q/(2πlε0)ln(r). l is for the length of the gaussian surface which i don't know and i don't see anywhere that I can cancel l . Besides, I have no idea where I can use that dielectric strength since electric field vary.

 

[haruspex]

Q/(2πlε0)ln(r). l is for the length of the gaussian surface which i don't know and i don't see anywhere that I can cancel l

You are asked for the maximum voltage, not the maximum charge. The longer the tube and wire, the greater the total charge.
Work with the charge per unit length. Relate that to the voltage and to the field (as function of r).

 

[xxwwyytt]

You are asked for the maximum voltage, not the maximum charge. The longer the tube and wire, the greater the total charge.
Work with the charge per unit length. Relate that to the voltage and to the field (as function of r).

I tried to relate to the E field by using Gauss's Law, and I get E = λ/2πrε0. Integrate E, I get V= (λ/2πε0)ln(r), and then I don't know where I should integrate from. Should I integrate from 1.5e-2m to 0.1e-3m or do I get r by plugging the dielectric strength into the equation of E?

 

[haruspex]

Should I integrate from 1.5e-2m to 0.1e-3m

If you do that, what exactly does the resulting equation tell you?

do I get r by plugging the dielectric strength into the equation of E?

If you do that, what exactly does the resulting equation tell you?

 

[xxwwyytt]

If you do that, what exactly does the resulting equation tell you?

If you do that, what exactly does the resulting equation tell you?

Thanks! I figure out that I should get Qmax first and then plug in 1.5e-2 and 0.1e-3 to find the Vmax.

 

[haruspex]

Thanks! I figure out that I should get Qmax first and then plug in 1.5e-2 and 0.1e-3 to find the Vmax.

Perhaps, but how do you plan to find Qmax?