Two identical charges of the same sign are separated by a distance of 2
d. The magnitude of the
electric field at the points along the perpendicular bisector of the line joining the two charges is
given by
\[
\begin{gather}
E=\frac{1}{4\pi \epsilon_0}\frac{2qy}{\left(a^2+y^2\right)^{3/2}}
\end{gather}
\]
Determine:
a) The points along the
y-axis, for which the magnitude of the electric field assumes its maximum
value;
b) The magnitude of the maximum electric field.
Electric Field of a Continuous Charge Distribution
A ring of radius a carries a uniformly distributed electric charge Q. Calculate the
electric field vector at a point P on the symmetry axis perpendicular to the plane of the ring
at a distance z from its center.
A ring of radius
a is uniformly charged with a charge
Q. The electric field produced by this
ring at points on the axis of symmetry perpendicular to the plane of the ring at a distance
z is
given, in magnitude, by
\[
\begin{gather}
E=\frac{1}{4\pi \epsilon_{0}}\frac{Qz}{\left(a^{2}+z^{2}\right)^{3/2}}
\end{gather}
\]
Determine:
a) For what values of z is the electric field maximum?
b) What is this maximum value.
Two concentric rings are located on the same plane. The ring of radius
R1 has a charge
Q1, and the ring of radius
R2 has a charge
Q2. The
electric field vector produced by a ring of radius
r at a distance
z from the center is
given by
\[
\begin{gather}
\mathbf{E}=\frac{1}{4\pi \epsilon_0}\frac{Qz}{\left(r^2+z^2\right)^{3/2}}\;\mathbf{k}
\end{gather}
\]
Determine the electric field vector:
a) In the common center of the two rings;
b) At a point located at a distance z, much greater than
R1 and
R2.
An arc of a circle of radius a and central angle θ0 carries an electric charge
Q uniformly distributed along the arc. Determine:
a) The electric field vector, at the points of the line passing through the center of the arc and is
perpendicular to the plane containing the arc;
b) The electric field vector in the center of curvature of the arc;
c) The electric field vector when the central angle tends to zero.
A ring of radius a carries a uniformly distributed electric charge q1 in one of
the halves and q2 in another half. Calculate the electric field vector at a point
P on the symmetry axis perpendicular to the plane of the ring at a distance z from its
center.