VDU_Physics_3_004 Solved University-Physics Problems

Physics 3:  Introduction to Electricity & Magnetism

<<<  Electric Field  >>>

Javier Montenegro Joo

jmj@VirtualDynamicsSoft.com <<<  Electric Field  >>>

(1) Four equal charges of magnitude Qo are placed equidistant from each other on the edge of a circumference of radius R.  Calculate the resultant electric field at a point on the perpendicular axis of the circumference and at a distance R from the plane of the circumference.

Solution.-

First, find only the resultant electric field from a pair of symmetrical charges, for example those two on the vertical, at the end the total electric field due to the four charges will be the double of this result. The sketch shows the situation for the vertical pair of charges, it can be seen that the vertical components of the electric fields (E1Y and E2Y) of these two charges are equal and cancel each other, hence the resultant field of this pair is the sum of their horizontal components, which are also equal (E1x = E2x): Physics Virtual Lab (PVL,  200 Physics simulation modules):

Electrically-charged particle deflection by an electric field: the user sets the electric field intensity, the mass of the particle, its electric charge and its initial velocity, the module executes the simulation. The PVL is a highly interactive virtual lab of physics created by Javier Montenegro Joo, computational & simulational physicist

To know more about the PVL click:     http://www.virtualdynamicssoft.com

(2) In the fig., a long and straight wire having a linear charge density is placed in front, parallel and very close to a sheet of paper having a surface charge density. The short distance from wire to paper is ‘b’. Determine the distance ‘r’ measured from the wire, where a point charge Q does not experiment any force. Solution.-

An electric charge placed between the wire and the plane will not experiment any force, if the electric field of the wire equals that of the plane: (3) In the sketch, a charge Q is uniformly distributed along a circumference arc of radius b, which spans an angle from 45to 135o.  Determine the electric field at the origin of coordinates. Solution.-

In order to solve this problem, consider a differential element of charge at the arc, sketch its electric field vector at the origin of coordinates and show its vector components along the x and y directions. Then repeat the procedure for another differential element of charge at a symmetric position in the arc. The result is that the components along the x-direction cancel each other, while the components along the y-direction add together producing the resultant electric field, which is downwards. El Asombroso Arte Algoritmico:

(4)  Four electric charges,   5Qo ,   -4Qo ,  6Q and  Q,  are placed each in a corner of a square of side 2L.  Determine:  The charge Q, so that no electric field is detected at the center of the square. Solution.- (5)  In the figure, a wire has been bent so that it has the shape of a semi-circumference of radius R, then the wire has been loaded with an electric charge and, this distributes as a linear density  L=LoSin(Theta). Determine the electric field at the origin of coordinates. Solution.-

Integrating the electric field between 0o  and 180o  (from top to bottom of the sketched arc) generates a weird result, which is neither in agreement with the symmetry of the Sine function between the mentioned points, nor with the symmetry of the arc.

The shape of the Sine function between 0o and 180, shows that the function decreases at both extremes, this is, top and bottom of the arc and, it takes a maximum at 90o , which in this sketch is around the x-axis.   Thus by an analysis of symmetry the resulting electric field must be in the x-direction.

Consider a differential charge dQ in upper region of the arc, sketch its Ex and Ey at the origin of coordinates. Then do the same for a symmetrical dQ in the lower region of the arc, it can be seen that the vertical components Ey of both dQ cancel each other and only survive the Ex components of both dQ, which must be integrated. The sketch shows an Electric Dipole, this is, a pair of charges of equal magnitude and different signs, separated by a 2b distance.  Find the resultant electric field at a point situated in x, at the right-hand side of the negative charge. Solution.-

Both electric charges have the same magnitude, but the negative one exerts more influence on point X than the positive charge, because the negative charge is closer to X, as a consequence the resultant electric field will point towards the negative charge: Physics Virtual Lab (PVL,  200 Physics simulation modules)

Optics Simulators The PVL is a highly interactive virtual lab of physics created by Javier Montenegro Joo, computational & simulational physicist.

To know more about the PVL click:     http://www.virtualdynamicssoft.com

 A small plastic ball having an electric charge equivalent to 18 elementary electric charges is placed in the origin of coordinates of a Cartesian system. (a) Determine the total charge in the ball (b) Calculate the electric field at x = 10 cm of the ball (c) Find the force perceived by a 5c charge placed at  y = - 10 cm.

Solution.-

The electric field due to the charge at the origin of coordinates goes outward in all directions and, its magnitude is the same at equal distances in all directions. Javier Montenegro Joo's  Chaos Virtual Lab: Experimental Findings in the Chaotic Nonlinear Damped and Forced Oscillator

 The electric field at a radial distance r from a copper-made and electrically charged sphere of radius R is For the sphere determine its:  (a) Surface charge (b) Interior charge (c) Total charge.

Solution.-

Since the sphere is metallic all the electric charge is on its surface and there is no charge in its interior. At the surface of the sphere the radius is r = R and the electric field is: Five charges Qo, 2Qo, 3Qo, 4Qo and 5Qo are equidistantly placed on a semi-circumference of radius R, such as it is shown in the figure.  Find the resultant electric field at the center of the circumference.  Compute the force experienced by a charge Q placed at the center of the circumference.  Four electric charges Qo, 2Qo, 3Qo and 4Qo, are sequentially placed each at a corner of a square of side 2L. Compute the resultant electric field at the center of the square. Find the resultant force experienced by a charge Q placed at the center of the square.

Solution.-

As it can be seen in the sketch, each one of the four charges at the corners of the square generates an electric field vector (E1, E2, E3 and E4) at the center of the square. After an algebraic sum of fields, only two diagonal vectors remain and, finally these add up to generate the vertically upwards resultant electric field. 