#### Question

It is desired to measure the magnitude of field between the poles of a powerful loud speaker magnet. A small flat search coil of area 2 cm^{2} with 25 closely wound turns, is positioned normal to the field direction, and then quickly snatched out of the field region. Equivalently, one can give it a quick 90° turn to bring its plane parallel to the field direction). The total charge flown in the coil (measured by a ballistic galvanometer connected to coil) is 7.5 mC. The combined resistance of the coil and the galvanometer is 0.50 Ω. Estimate the field strength of magnet.

#### Solution

Area of the small flat search coil, *A* = 2 cm^{2 }= 2 × 10^{−4} m^{2}

Number of turns on the coil, *N* = 25

Total charge flowing in the coil, *Q* = 7.5 mC = 7.5 × 10^{−3} C

Total resistance of the coil and galvanometer, *R* = 0.50 Ω

Induced current in the coil,

`I=("Induced emf(e)")/R` ...(1)

Induced emf is given as:

`e=-N(dphi)/(dt)` ...(2)

Where,

`dphi` = Charge in flux

Combining equations (1) and (2), we get

`I=-(N(dphi)/(dt))/(R)`

`Idt=-N/Rdphi`

Initial flux through the coil, `phi_i `= *BA*

Where,

*B* = Magnetic field strength

Final flux through the coil, `phi_f=0`

Integrating equation (3) on both sides, we have

`intIdt=-N/Rint_(phi_i)^(phi_f)dphi`

But total Charge Q=`intIdt`

`therefore Q=-N/R(phi_f-phi_i)=-N/R(-phi_i)=+(Nphi_i)/R`

`Q=(NBA)/R`

`therefore B=(QR)/(NA)`

`=(7.5xx10^-3xx0.5)/(25xx2xx10^-4)=0.75 T`

Hence, the field strength of the magnet is 0.75 T.