A current of `10 A` flows around a closed path in a circuit which is in horizontal plane as shown in the figure . The circuit consists of eight alternating arcs of ` radii r_(1) = 0.08 m and r_(1) = 0.12 m. Each arc subtends the same at he center.
(a) Find the magnetixc field produced by this circuit at the center.
(b) An infinitely long straight wire carrying a current of `10 A` is passing through the center of the above circuit vertically with the direction of the current being into the plane of the circuit . What is the force acting on the arc `AC` and the straight segment ` CD` due to the current at the center?

Answer

(i) 6.54 (ii) (0, 8.1)

Explanation

(i) 6.54 (ii) (0,8.1)
(i) Given `i=10A,r_(1)=0.08m` and `r_(2)=0.12m`. Straight portion i.e. CD etc., will produce zero magnetic field at the centre. Rest eight arcs will produce the magnetic field at the centre in the same direction i.e., perpendicular to the paper outwards or vertically upwards and its magnitude is
`B=B_("innerarcs")+B_("outerarcs")`
`=1/2{(mu_(0)i)/(2r_(1))}+1/2{(mu_(0)i)/(2r_(2))}=((mu_(0))/(4pi))(pii)((r_(1)+r_(2))/(r_(1)r_(2)))`
Substituting the values we have
`B=((10^(-7))(3.14)(10)(0.08+0.12))/((0.08xx0.12))`
`B=6.54xx10^(-5)T`
(vertically upward or outward normal to the paper)
(ii) Force on AC
Force on circular portions of the circuit i.e., AC etc, due to the wire at the centre will be zero because magnetic field due to the central wire at these arcs will be tangential `theta=180^(@)`.
Force on CD
Current in central wire is also `i=10A`. Magnetic field at distance x due to central wire
`B=(mu_(0))/(2pi).i/x :.` Magnetic force on element dx due to this magnetic field
`dF=(i)((mu_(0))/(2pi). i/x). dx=((mu_(0))/(2pi)i^(2)(dx)/x, (F=ilBsin 90^(@))`
Therefore net force on CD is
`F=int_(x=r_(1))^(x=r_(2))dF=(mu_(0)i^(2))/(2pi)int_(0.08)^(0.12) (dx)/x=(mu_(0))/(2pi)i^(2) In (3/2)`
Substituting the values `F=(2xx10^(-7))(10)^(2)In(1.5)` or `F=8.1xx10^(-6)N` (in wards)
Force on wire at the centre, Net on wire at the centre
Net magnetic field at the centre due to the circuit is in vertical direction and current in the wire in centre is also in vertical direction. Therefore, net force on the wire at the centre will be zero. `(theta=180^(@))`. Hence,
(i) force acting on the wire at the centre is zero (ii) force on arc `AC=0`
(iii) force on segment CD is `8.1xx10^(-6)N` (inwards)