Question

There are a lot of numbers in this problem. Just about the only way to get it right is to work out each step symbolically first, and then plug numbers into the final symbolic result. 

Two coils of wire are aligned with their axes along the z-axis, as shown in the diagram. Coil 1 is connected to a power supply and conventional current flows counter-clockwise through coil 1, as seen from the location of coil 2. Coil 2 is connected to a voltmeter. The distance between the centers of the coils is 0.16 m. 

Coil 1 has N1 = 510 turns of wire, and its radius is R1 = 0.09 m. The current through coil 1 is changing with time. At t=0 s, the current through coil 1 is lo = 15 A. At t=0.4 s, the current through coil 1 is 10.4 = 6 A.

 Coil 2 has N2 = 260 turns of wire, and its radius is R2 = 0.05 m. 

Answer 1

given

z = 0.16 m

N₁ = 510 turns

N₂ = 260 turns

R₁ = 0.09 m

R₂ = 0.05 m

i_o = 15 A

i_0.4 = 6 A

a )

$\phi$_t=0 = 2 $\mu$ _o N₁ i_o A₁ A₂ / ( z² + R₁² )^3/2

= 2 $\mu$ _o N₁ i_o$\pi$² R₁² R₂² / ( z² + R₁² )^3/2

= 2 x 4 x 3.14 x 10^-7 x 510 x 15 x 3.14² x 0.09² x 0.05² / ( 0.16² + 0.09² )^3/2

$\phi$_t=0 = 6.201 x 10^-4 T-m²

b )

$\phi$_t=0.4 = 2 $\mu$ _o N₁ i_o.4 A₁ A₂ / ( z² + R₁² )^3/2

= 2 $\mu$ _o N₁ i_o$\pi$² R₁² R₂² / ( z² + R₁² )^3/2

= 2 x 4 x 3.14 x 10^-7 x 510 x 6 x 3.14² x 0.09² x 0.05² / ( 0.16² + 0.09² )^3/2

$\phi$_t=0.4 = 2.48 x 10^-4 T-m²

c )

$\varepsilon$ = | d$\phi$/dt |

= | ( 2.48 x 10^-4 - 6.201 x 10^-4 ) / 0.4 |

$\varepsilon$ = 9.325 x 10^-4 V

d )

V = N₂$\varepsilon$

= 260 x 9.325 x 10^-4

V = 0.2424 V

e )

E_NC = V / L₂

= V / 2 $\pi$ R₂ N₂

= 0.2424 / 2 x 3.14 x 0.05 x 260

E_NC = 2.969 x 10^-3 V/m

so first two selected options are correct