Fisicaexe - Solved Problems in Physics

Coulomb's Law and Electric Field

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Coulomb's Law

We want to share a charge Q between two bodies. One of the bodies receives a charge q₁ and the other a charge q₂. The splitting of the charges is made in such a way that if you have q₁+q₂=Q. Determine the ratio between the charges so that the Coulomb's force of repulsion between q₁ and q₂ is maximum for any distance between the charges.

Solution

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.

Solution

Electric Field of a Discrete Charge Distribution

Two equal charges of the same sign are separated by a distance of 2d. 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.

Solution

Suggestion: compare with the calculation of the magnitude of the electric field

Two identical charges of the same sign are separated by a distance of 2d. 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.

Solution

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

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.

Solution

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.

Solution

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.

Solution

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.

Solution

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.

Solution

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