Answer: The magnetic field points to the west.
Explanation:
If we take the plane as:
North = positive y-axis
East = positive x-axis.
We have that the electron is moving south, so the velocity of the electron can be written in vector form as:
V = (0, -v, 0)
Now, when the electron interacts with the magnetic field, the electron moves upwards.
We know that the interaction between an electron and a magnetic field is:
F = q*VxB
So we have that this force acts on the z-direction.
Now, to solve this we can use the righ hand rule.
First, we point wit our hand to the direction of the velocity (negative y-axis) now, we want that our thumb points up (the direction of the force) so the side where our palm faces is the direction of the field B.
But, remember that an electron has a negative charge, so the actual equation is:
F = -q*VxB
So the magnetic field actually points in the opposite direction of our palm, to the west.
Now, we can also solve it mathematically as:
F = (0, 0, f) = -q*(0, -v,0)x(a, b, c)
where B = (a, b, c) is the vector of the magnetic field.
(0, 0, f) = -q*(-v*b -0, -0 + 0, 0 -(- a*v)) = -q*(-v*b, 0, a*v)
then we have that b must be equal to zero, and that:
f = -q*a*v
and f is positive, then we have:
a = -(f/q*v)
then the vector of the magnetic field is:
B = (-(f/q*v), 0, 0)
so it points in the negative x-axis, that is the West, as we found earlier.
