Coulomb's Law and Electric Field

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We want to share a charge *Q* between two bodies. One of the bodies receives a charge
*q*_{1} and the other a charge *q*_{2}. The splitting of the charges is made in
such a way that if you have *q*_{1}+*q*_{2}=*Q*. Determine the ratio between
the charges so that the Coulomb's force of repulsion between *q*_{1} and *q*_{2}
is maximum for any distance between the charges.

Two charges of the same magnitude and opposite signs are fixed on a horizontal line at a distance *d*
from each other. A sphere, with mass *m*, carries an electric charge, attached to a wire is approximate,
first of one of the charges until it is in equilibrium exactly on it at a height *d*. Following, the
wire is moved toward the second charge until the charge is in equilibrium over the second charge. Determine
the angles of deviation in both cases, knowing that in the first case the deviation angle is twice higher
than the deviation angle in the second case.

Two equal charges of the same sign are separated by a distance of 2*d*. Calculate the electric field
vector at the points along the perpendicular bisector of the line joining the two charges. Check the
solution for points far away from the center of the dipole.

Two equal charges of the same sign are separated by a distance of 2*d*. Calculate the electric field
vector at the points along the perpendicular bisector of the line joining the two charges. Check the
solution for points far from the charges.

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

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

\[
\begin{gather}
E=\frac{1}{4\pi \epsilon_0}\frac{2qy}{\left(a^2+y^2\right)^{3/2}}
\end{gather}
\]

Determine:

a) The points along the

b) The magnitude of the maximum electric field.

Solution

**Suggestion:** compare with the maximum points of the
electric field
*of a loop charged with charge Q.*

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

Determine:

a) For what values of z is the electric field maximum?

b) What is this maximum value.

\[
\begin{gather}
E=\frac{1}{4\pi \epsilon_{0}}\frac{Qz}{\left(a^{2}+z^{2}\right)^{3/2}}
\end{gather}
\]

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 *R*_{1} has a charge
*Q*_{1}, and the ring of radius *R*_{2} has a charge *Q*_{2}. The
electric field vector produced by a ring of radius *r* at a distance *z* from the center is
given by

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*R*_{1} and *R*_{2}.

\[
\begin{gather}
\mathbf{E}=\frac{1}{4\pi \epsilon_0}\frac{Qz}{\left(r^2+z^2\right)^{3/2}}\;\mathbf{k}
\end{gather}
\]

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

An arc of a circle of radius *a* and central angle θ_{0} 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) 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 *q*_{1} in one of
the halves and *q*_{2} 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.

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Fisicaexe - Physics Solved Problems by Elcio Brandani Mondadori is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .