Question

Carefully measure the magnetic field at the end of the solenoid. How does it compare to the value at the center of the solenoid? Try to prove what the value at the end should be.

Answer 1

Start

Magnetic field at the end of a solenoid became half than field at the centre.

Bo = 2 Be.

When the value Bo of the field strength in the center is compared with the value Be of the field strength at one end, the Bo/Be ratio is equal to the ratio of the geometric factors. This ratio is the square root (sqrt) of the quotient

(1 +4p^2)/(1 +p^2)

where the parameter p is the length/diameter ratio of the solenoid, i.e.

p = L/(2R)

Then, the following expresses the EXACT calculation:

Bo/Be = sqrt [ (1 +4p^2) / (1 +p^2) ]

As you can see, if the solenoid is VERY long, p is much greater than 1, and then the square root is approximately 2:

Bo/Be = sqrt [ (1 +4p^2) / (1 +p^2) ]

sqrt [ (4p^2) / (p^2) ] = sqrt ( 4 )

= 2

End