A vacuum tube diode consists of concentric cylindrical electrodes, the negative

Question

A vacuum tube diode consists of concentric cylindrical electrodes, the negative cathode and the positive anode. Because of the accumulation of charge near the cathode, the electric potential between the electrodes is not a linear function of the position, even with planar geometry, but is given by V(z) = Cz4/3 where z is the distance from the cathode and C is a constant characteristic of a particular diode and operating conditions. Assume that the distance between the cathode and anode is 11.0 mm and the potential difference between electrodes is 220 V

Explanation / Answer

V(x) = C x4/3

given , V(0.011) = 220

C (0.011)4/3 = 220

C = 89928.9

Potential energy is given as

U(x) = - q V(x) = - q C x4/3

force is given as

F(x) = - dU(x) /dx

F(x) = - (d/dx) ( - q C x4/3 )

F(x) = (4q/3) C x1/3

at halfway x = 5.5 mm = 0.0055 m

F(0.0055) = (4(1.6 x 10-19)/3) (89928.9) (0.0055)1/3 = 3.4 x 10-15 N

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