A magnetic field of `100G(1G=10^-4T)` is required which is uniform in a region of linear dimension about `10cm` and area of crossection about `10^-3m^2`. The maximum current carrying capacity of a given coil of wire is `15A` and the number of turns per unit length that can be wound round a core is at most `1000turns m^-1`. Suggest some appropriate design particulars of a solenoid for the required purpose. Assume the core is not ferromagnetic.

Explanation

The magnitude of magnetic field produced inside a solenoid of number of turns per unit length and current I flowing through the coil is given by
` B = mu_(0) n I`
Here, `B = 100 xx 10^(-4) T`
` rArr " " nI = B/mu_(0) = (100 xx 10^(-4))/(4 pi xx 10^(-7)) = 7,961 approx 8,000 ` A/m
It is given that maximum current capacity of coil, ` I = 15 A`
And maximum number of turns per unit length that can be wound round the core, `n = 1000` turns/m So, if we take `n = 1,000 " turns/m," I = 8 A`
If we take ` I = 10 A` ,
` n = B/(mu_(0)I) = (8,000)/10 = 800 ` turns/m
For any value of `I gt 10` A, n will become fractional and for `I lt 8 `A, n will become greater than 1,000 . So, `I = 10 ` A and ` n = 8000` seems to be an optimal value for the required design.
To obtain a uniform magnetic field of about 10 cm in length inside a solenoid, the core length should be nearly 5 times the length of uniform magnetic field region (which can be obtained in the centre region of the core). So, we take core length ` = 10 xx 5 = 50 ` cm. To have a uniform magnetic field over a cross section of `10^(-3) m^(2)` , the cross section on which the wire can be wound is five times the given area i.e.,`5 xx 10^(-3) m^(2)` Let r be the radius of care. Then
` pi r^(2) = 5 xx 10^(-3)`
`r = sqrt((5 xx 10^(-3))/pi) = 0.039` m