Introductory Physics Volume Two

4.6 More Examples 89
-e
F E
v
F B
= -e(v x B)
Again, since the electron’s charge is negative, the magnetic force will
point in the direction opposite the cross-product ⃗v × ⃗ B. So, ⃗ B must
point into the page. The magnitude of B can be determined since the
net force is to be zero:
F B = F E
−→ evB = eE
Or, solving for B:
B = E v
For a parallel plate capacitor: E = σ/ɛ 0 , so
B =
σ Q
ɛ 0 v = A
ɛ 0 v
7.5 × 10 −6 C
−→ B =
(0.2m) 2 (8.85 × 10 −12 C 2
Nm
)(4 × 10 6 m
2 s ) = 5.3T
So, ⃗ B must be perpendicular to the electric field, pointing into the page
with a magnitude of 5.3T for the electon to have no net force on it.
Example
A proton is injected from a region with no magnetic field into a region
with a strip of uniform magnetic field that is 10cm wide:
+e
v = ?
L = 10cm
B = 0.3T
The magnetic field points into the page and has a magnitude of 0.3T.
What is the maximum velocity the proton can have such that it does
not make it across the 10cm magnetic-field region and exit through the
other side?
Upon entering the magnetic field, the proton will experience a force
perpendicular the velocity, which will cause it to follow a circular path: