How Do You Calculate Electric Field Strength in a Geiger Counter?

[thehigaman]

I've been working on a problem but I don't understand how to solve it.

A Geiger counter detects radiation such as alpha particles by using the fact that the radiation ionizes the air along its path. A thin wire lies on the axis of a hollow metal cylinder and is insulated from it. A large potential difference is established between the wire and the outer cylinder, with the wire at higher potential; this sets up a strong electric field directed radially outward. When ionizing radiation enters the device, it ionizes a few air molecules. The free electrons produced are accelerated by the electric field toward the wire and, on the way there, ionize many more air molecules. Thus a current pulse is produced that can be detected by appropriate electronic circuitry and converted to an audible "click". The tube of a Geiger counter has a long, hollow, metal cylinder 2.00 cm in diameter. Along the axis of the tube is a wire 0.127 mm in diameter running its full length. When the tube is operating, a voltage of 850 V is applied between the two conductors.

Find the electric field strength at the outer surface of the wire & the electric field strength at the inner surface of the cylinder.

I was trying to use this eqn: E(r)=V/ln(b/a)*(1/r) where b=radius of the cylinder and a=radius of wire.

Any help would be appreciated. Thank you.

 

[vanesch]

Find the electric field strength at the outer surface of the wire & the electric field strength at the inner surface of the cylinder.

I was trying to use this eqn: E(r)=V/ln(b/a)*(1/r) where b=radius of the cylinder and a=radius of wire.

Any help would be appreciated. Thank you.

The form of the E-field is correct (I didn't check the ln(b/a) factor ; it must be something of the kind ; you can verify it by integrating the E-field along a radial line from the surface of the wire to the radius of the tube). So your formula gives you the E-field for an arbitrary distance r from the center of the wire. They ask you to give the value of the E-field for two particular places ; why can't you just put in the numbers for those places (particular values of r) ?

EDIT: a remark to the problem of general nature: you need to use special gas mixtures. A geiger counter doesn't work in AIR The reason is that oxygen is way too electronegative, and has "eaten" all electrons drifting before they reach the anode wire.

 

[quicknote]

I can understand how this problem may be confusing. It involves several concepts such as electric fields, potential difference, and ionizing radiation. To solve this problem, we can use the equation you mentioned, E(r) = V/ln(b/a)*(1/r), where b is the radius of the cylinder and a is the radius of the wire. However, it is important to note that this equation applies to a cylindrical capacitor, where the electric field is uniform between the two conductors.

In this case, we are dealing with a non-uniform electric field as the wire is not parallel to the outer cylinder. To solve for the electric field strength at the outer surface of the wire and the inner surface of the cylinder, we need to use the equation for electric field strength due to a point charge, E = kQ/r^2, where k is the Coulomb's constant, Q is the charge, and r is the distance from the charge.

In this problem, the wire acts as a point charge due to the high voltage applied to it. Therefore, we can use this equation to find the electric field strength at the outer surface of the wire by considering the wire as a point charge at its center. Similarly, we can find the electric field strength at the inner surface of the cylinder by considering the charge on the wire as a point charge at the inner surface of the cylinder.

I would suggest using this approach to solve the problem and ensure that the units are consistent. I hope this helps you understand the problem better. If you need further assistance, please don't hesitate to reach out. I am always happy to help with any scientific problems.