Question

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

Answer 1

Given, Magnetic field strength,

$B=100G=100\times {10}^{-4}$ T
maximum number of turns per unit length,
$n_{max}=1000\text{turns}/m$
Current carrying capacity of the coil

$I_{max}=15A$
Area of cross-section $A\approx {10}^{-3}{m}^2$
Permeability of free space,
${\mu }_0=4\pi \times {10}^{-7}Tm{A}^{-1}$
Magnetic field is given as
$B={\mu }_0\frac{NI}{l}={\mu }_{0}nI$

$\Rightarrow nI=\frac{B}{{\mu }_0}=\frac{100\times {10}^{-4}}{4\pi \times {10}^{-7}}$
$nI=7961$
Then for $n_{max}=1000\text{turns/m}$
$I=\frac{7961}{1000}=7.961A$

Therefore, $7.961A$ should be the current parameter to produce magnetic field of $100G$ keeping number of turns per unit length as $1000\text{turns/m}$.
The current can be increase upto $15A$ by decreasing the number of turns per unit length. Hence the parameters are not unique ans some adjustments within limits is possible.

Final Answer: $7.961A$ should be the current parameter to produce magnetic field of $100G$ keeping number of turns per unit length as $1000turns/m$.

The parameters are not unique and some adjustments within limits is possible.