HW #2 – Solutions
[E-Field Lines & Particle Paths]
Remember that electric field lines are constructed so that the tangent to the field line at any point gives the direction of the
electric field at that point.
The electric force is proportional to the electric field, so the force vector (on a positive charge)
will point in the same direction as the electric field.
Since the acceleration is proportional to the force, the acceleration
vector will point along the tangent to the electric field line.
Now let’s consider the particle’s trajectory.
If we look at the path the particle follows, the velocity vector at a point on the
path is tangent to the path at that point.
So the shape of the electric field lines tell us about the acceleration, and the shape of
the particle’s path tells us about the velocity.
In Fig.21.29a, the field lines are straight lines so the force is always in a straight line and velocity and acceleration are
always in the same direction. The particle moves in a straight line along a field line, with increasing speed.
In Fig.21.29b, the field lines are curved. Suppose the charged particle followed the path of one of these curved field lines.
Remember that, for the particle to follow a curved path, its acceleration must have a component perpendicular to the path.
This is not possible if the particle is following a field line, since the electric field (and, hence, the acceleration) must be
tangent to the field line.
So the path the particle follows must be different.
As the particle moves its velocity and
acceleration are not in the same direction and the trajectory does not follow a field line.
In conclusion, two-dimensional motion the velocity is always tangent to the trajectory but the velocity is not always in the direction of
the net force on the particle.
[Infinite Line Charge E-Field Lines]
We use symmetry to deduce the nature of the field lines.
The only distinguishable direction is toward the line or away
from the line, so the electric field lines are perpendicular to the
line of charge, as shown in the figures to the right.
From the figures above, we see that
the main difference between the electric field from a point charge and the
electric field of a charged line is that the field lines from a point charge spread out in all three dimensions,
while the field lines from a line of charge spread out in only two dimensions—the dimensions perpendicular
to the line. The magnitude of the electric field is inversely proportional to the spacing of the field lines, so
spreading out in fewer dimensions means the field falls off more slowly. To see how this spacing between
field lines changes with
, the distance from the line, imagine that the line of charge is surrounded by a
cylinder of radius
centered on the charged line. Since all of the spreading is done in the directions perpendicular to the line,
it is sufficient to look at the field in a plane perpendicular to the line.
This is shown on the figure to the right.
One sees that,
on this plane, the line of charge appears as a point, the cylinder appears as a circle, and the field lines are radial lines.