An automobile has a vertical radio antenna 110 m long The au

An automobile has a vertical radio antenna 1.10 m long. The automobile travels at 63.0 km/h on a horizontal road where Earth\'s magnetic field is 50.0 µT, directed toward the north and downward at an angle of 65.0° below the horizontal.

Calculate the magnitude of this induced emf in mV.

Solution

The induced emf of a conductor L moving with velocity V in a magnetic field B is given by the formula;

E = (V cross B) dot ( L )

Where V , B & L are vectors.

. Let x be the horizontal East/West direction. Let y be the North/South direction and Let z be perpendicular to the plane or the ground.

The unit vectors in these directions are (i, j, k )

Then from the given information your vectors are;

V = (Vx)i and L = (Lz)k and B = (By)j - (Bz)k (minus for below horizontal)

Now (V cross B) = (Vx)(By)(i cross j) - (Vx)(Bz)(i cross k)

= (Vx)(By)k + (Vx)(Bz)j

Now Dot this into (L)k and only the first term is non zero;

(V cross B) dot (L) = (Vx)(By)(Lz)

Where; Vx = 63 km/h = 17.5 m/s

Lz = 1.1 m

By = (B)Sin(65) = (50 x 10^-6)(.88) = 45.3 x 10^-6 T

E = (17.5)(1.1)(45.3 x 10^-6) = .000872 volts

I hope your familiar with vector products between unit vectors.