An electron, moving with a speed, 6.8 km/s, enters a magnetic field that has a strength of 6.9 T. The magnetic force on the electron is perpendicular to the electron's velocity. Determine the radius of the resultant path of the electron.

centripetal force=magnetic force

mv^2/r=Bqv
solve for r.

To determine the radius of the resultant path of the electron, we can use the formula for the magnetic force on a charged particle moving in a magnetic field:

F = q * v * B * sin(theta),

where
F is the magnetic force,
q is the charge of the electron,
v is the velocity of the electron,
B is the magnetic field strength,
and theta is the angle between the velocity and the magnetic field.

In this case, the magnetic force is perpendicular to the electron's velocity, so the angle theta is 90 degrees, which means sin(theta) = 1.

The charge of an electron is q = -1.6 x 10^(-19) C (Coulombs).
The velocity of the electron is v = 6.8 km/s = 6.8 x 10^3 m/s.
The magnetic field strength is B = 6.9 T (Tesla).

Plugging these values into the formula for the magnetic force:

F = (-1.6 x 10^(-19) C) * (6.8 x 10^3 m/s) * (6.9 T) * 1.

Now we can use another formula to calculate the radius of the resultant path, known as the centripetal force formula. The centripetal force on the electron is provided by the magnetic force:

F = m * a,

where
m is the mass of the electron,
a is the centripetal acceleration of the electron.

The mass of an electron is m = 9.1 x 10^(-31) kg.

Since the centripetal acceleration is given by a = v^2 / r, where r is the radius of the resultant path, we can rewrite the centripetal force formula as:

F = m * (v^2 / r).

Setting F equal to the magnetic force, we can write:

(m * (v^2 / r)) = (-1.6 x 10^(-19) C) * (6.8 x 10^3 m/s) * (6.9 T) * 1.

Now we can solve for r.

First, multiply both sides by r:

m * v^2 = (-1.6 x 10^(-19) C) * (6.8 x 10^3 m/s) * (6.9 T) * r.

Then, divide both sides by (m * v^2):

r = [(-1.6 x 10^(-19) C) * (6.8 x 10^3 m/s) * (6.9 T)] / [ m * v^2 ].

Now plug in the known values:

r = [(-1.6 x 10^(-19) C) * (6.8 x 10^3 m/s) * (6.9 T)] / [ (9.1 x 10^(-31) kg) * (6.8 x 10^3 m/s)^2 ].

After performing the calculations, you will obtain the value for the radius of the resultant path of the electron.