Deck 3: Static Electric Fields

Question 1

Two point charges, Q 1 and Q 2 , are located at (0, 5, 1) and (0, 2, 6), respectively. Find the relation between Q 1 and Q 2 such that the total force on a test charge at the point P(0, 2, 3) will have
a) no y -component, and
b) no z -component.

Answer 1

The point charge
is located at
.
The point charge
is located at
.
The test charge
is placed at
.
Determine the force on
due to
Determine the force on
due to
.
Determine the total force on
due to
and
(a)
If the total force on test charge will have no y-component, then the coefficient of
must be zero.
Thus the relation between
and
if the total force on test charge will have no y-component is
.
(b)
If the total force on test charge will have no z-component, then the coefficient of
must be zero.
Thus the relation between
and
if the total force on test charge will have no y-component is
.

Question 2

Describe the ways in which the electric field intensity of an infinitely long, straight line charge of uniform density varies with distance.

Answer 2

Write an expression for the intensity of electric field due an infinitely long straight line charge.
Where,
is unit vector
is linear charge density
is permittivity of free space
r is the perpendicular distance from the line to that point where we have to find the electric field intensity
The magnitude of intensity of electric field is inversely proportional to the distance that is as the distance increases from an infinitely long straight line charge, the magnitude of electric field intensity decreases.

Question 3

Assume the Earth to be a large conducting sphere (radius = 6.37 × 10 3 km) surrounded by air. Find its capacitance referring to infinity.

Answer 3

Write an expression for the capacitance of spherical capacitor.
…… (1)
Where,
is the radius of the Earth
is permittivity of free space
Substitute
for radius of the Earth and
for Coulomb's constant
in equation (1).
Thus, the required capacitance,
is
.

Question 4

Dielectric lenses can be used to collimate electromagnetic fields. In Fig. 3-30 the left surface of the lens is that of a circular cylinder, and the right surface is a plane. If E 1 at point P ( r o , 45°, z ) in region 1 is a r 5 a 3, what must be the dielectric constant of the lens in order that E 3 in region 3 is parallel to the x -axis
Fig A dielectric lens (Problem P.3-19).

Question 5

Write the expression for electrostatic energy in terms of E.

Question 6

What is the image of a spherical cloud of electrons with respect to an infinite conducting plane

Question 7

Write the integral form of the fundamental postulates of electrostatics in free space, and state their meaning in words.

Question 8

If the polystyrene of the coaxial cable in Example 3-12 is replaced by air, what would the maximum allowable working voltage of the cable be (Maintain the restriction that the maximum field intensity is not to exceed 25% of the dielectric strength of the insulating material.)

Question 9

The polarization vector in a dielectric sphere of radius b is P = a x P 0.
a) Determine the surface and volume charge densities.
b) Show that the total bound charge is zero.

Question 10

What are the general boundary conditions for E and D at an interface between two different dielectric media with dielectric constants r l and r 2

Question 11

For a coaxial capacitor of length L , use the method of virtual displacement to find the force between the inner conductor (radius a ) and the outer one (radius b ) that carry charges +Q and Q, respectively. The permittivity of the insulating material is .
Hint: First assume the inner and outer radii to be a and a + r, respectively; then differentiate with respect to r.

Question 12

An infinite line charge of 50(nC/m) lies 3(m) above the ground, which is at zero potential. Choosing the ground as the xy-plane and the line charge as parallel to the x-axis, find by the method of images the following:
a) E at (0, 4, 3) and
b) E and
at (0, 4, 0)

Question 13

Given E = a r (20/ r 2 ) (mV/m) in free space, find v at the point (3, 4, 1) (cm).

Question 14

Two infinitely long coaxial cylindrical surfaces, r = a and r = b ( b a ) carry surface charge densities sa and sb , respectively.
a) Determine E everywhere.
b) What must be the relation between a and b in order that E vanishes for r b

Question 15

What do we mean by a simple medium

Question 16

Find the potential distribution in the region R R o in Example 3-23.

Question 17

Discuss the meaning and use of the principle of virtual displacement.

Question 18

What is the image of an infinitely long line charge of density with respect to a parallel conducting circular cylinder

Question 19

Three point charges Q 1 = 9 ( C ), Q 2 = 4 ( C ) , and Q 3 = 36( C) are arranged on a straight line. The distance between Q 1 and Q 3 is 9 (cm), it is claimed that a location can be selected for Q 2 such that each charge will experience a zero force. Find this location.

Question 20

If the electric potential at a point is zero, does it follow that the electrical field intensity is also zero at that point Explain.

Question 21

Convert the kinetic energy of 2 (TeV) of the proton beam of a very powerful high-energy particle accelerator into joules.

Question 22

The space between a parallel-plate capacitor of area S is filled with a dielectric whose permittivity varies linearly from x at one plate ( y = 0) to 2 at the other plate ( y = d ). Neglecting fringing effect, find the capacitance.

Question 23

A parallel-plate capacitor of width w , length L , and separation d has a solid dielectric slab of permittivity in the space between the plates. The capacitor is charged to a voltage V o by a battery, as indicated in Fig. 3-32. Assuming that the dielectric slab is withdrawn to the position shown and the switch is opened, determine the force acting on the slab.

Question 24

The axis of a long two-mire parallel transmission line are 2(cm) apart. The wires have a radius 3(mm) and are maintained at potentials +100 (V) and -100 (V). Find
a) The location of the equivalent line charges relatives to the wire axes,
b) B) the equivalent line charge density of each wire, and
c) The electric field intensity at a point midway between the wires

Question 25

Explain why an irrotational field is also known as a conservative field.

Question 26

If it is desired that the working voltage of an air-filled coaxial cable having a radius r i = 2 (mm) for the inner conductor is to remain at 10 (kV) as in Example 3-9, what should r o be

Question 27

The axis of a long dielectric tube of inner radius r i and outer radius r o coincides with the z -axis. A polarization vector P = P 0 ( a x 3 x + a y 4 y ) exists in the dielectric.
a) Determine the surface and volume charge densities.
b) Show that the total bound charge is zero.

Question 28

What are the boundary conditions for electrostatic fields at an interface between a conductor and a dielectric with permittivity

Question 29

Write Poisson's and Laplace's equations in vector notation for a simple medium.

Question 30

Where is the zero-potential surface of the two-wire transmission line in Fig 3-29
Example

Figure Cross section of two-wire transmission line and equivalent line charges (Example 3-25)

Question 31

A positive charge Q is uniformly distributed on a very thin spherical shell of radius b in air. Find E everywhere. Plot |E| versus R.

Question 32

Determine the work done in carrying a +5( C) charge from P 1 (l, 2, 4) to P 2 ( 2 , 8, 4) in the field E = a x y + a y x
a) along the parabola y = 2 x 2 , and
b) along the straight line joining P 1 and P 2.

Question 33

Define electric displacement vector. What is its SI unit

Question 34

Find the total amount of charge induced on the surface of the infinite conducting plane in Example 3-24.

Question 35

The upper and lower conducting plates of a large parallel-plate capacitor are separated by a distance d and maintained at potentials V o and 0, respectively. A dielectric slab of dielectric constant 6.0 and uniform thickness 0.8 d is placed over the lower plate. Assuming negligible fringing effect, determine the following by solving Laplace's equation:
a) the potential and electric field distribution in the dielectric slab,
b) the potential and electric field distribution in the air space between the dielectric slab and the upper plate, and
c) the surface charge densities on the upper and lower plates.

Question 36

In Example 3-8 determine the position of the point P on the z -axis beyond which the disk may be regarded as a point charge if the error in the calculation of E is not more than 1%.

Question 37

If the electric field intensity at a point is zero, does it follow that the electric potential is also zero at that point Explain.

Question 38

Determine the amount of energy needed to arrange three point charges 1 ( µ C), 2 ( µ C), and 3 ( µ C) at the corners of an equilateral triangle of sides 10 (cm) in free space

Question 39

Assume that the outer conductor of the cylindrical capacitor in Example 3-16 is grounded and that the inner conductor is maintained at a potential V 0.
a) Find the electric field intensity, E ( a ), at the surface of the inner conductor.
b) With the inner radius, b, of the outer conductor fixed, find a so that E ( a ) is minimized.
c) Find this minimum E ( a ).
d) Determine the capacitance under the conditions of part (b).

Question 40

Write Poisson's and Laplace's equations in Cartesian coordinates for a simple medium.

Question 41

In what ways does the electric field intensity vary with distance for (a) a point charge (b) an electric dipole

Question 42

State and explain the boundary conditions that must be satisfied by the electric potential at an interface between perfect dielectrics with dielectric constants r l and r 2.

Question 43

A positive point charge Q is at the center of a spherical dielectric shell of an inner radius R i and an outer radius R o. The dielectric constant of the shell is r. Determine E, V, D, and P as functions of the radial distance R.

Question 44

What is the boundary condition for electrostatic potential at an interface between two different dielectric media

Question 45

Assume that the space between the inner and outer conductors of long coaxial cylindrical structure is filled with an electron cloud having volume density of charge v = A/r for a where a and b are, the radii of the inner and outer conductors, respectively. The inner conductor is maintained at a potential V o , and the outer conductor is grounded Determine the potential distribution in the region a by solving Poisson's equation.

Question 46

Determine the work done by the electric field E = a x x a y 2 y (V/m) in moving a unit positive charge from position P 1 ( 2, 0, 0) to position P 2 (5, 1, 3). The distances are in (m).

Question 47

Repeat problem P.3-10 if the field is E = a x y a y x.

Question 48

Define electric susceptibility. What is its unit

Question 49

A long power transmission line, 2 (cm) in radius, is parallel to and situated 10(m) above the ground. Assuming the ground to be an infinite flat conducting plane, find the capacitance per meter of the line with respect to the ground.

Question 50

If 2 U = 0, why does it not follow that U is identically zero

Question 51

Verify that the E field in Eq. (3-8) satisfies Eq. (3-4) and hence is conservative.

Question 52

A line charge of uniform charge density forms a circle of radius b that lies in the xy -plane in air with its center at the origin.
a) Find the electric field intensity E at the point (0, 0, h ).
b) At what value of h will E in part (a) be a maximum What is this maximum
c) Explain why E has a maximum at that location.

Question 53

Why are there no free charges in the interior of a good conductor under static conditions

Question 54

Two capacitors having capacitances 20 ( µ F) and 40 ( µF ) are connected in series across" a 60-(V) battery. Calculate the energy stored in each capacitor.

Question 55

The radius of the core and the inner radius of the outer conductor of a very long coaxial transmission line are r i and r o , respectively. The spa a between the conductors is filled with two coaxial layers of dielectrics. The dielectric constants of the dielectrics are r 1 for r i r b and r 2 for b o. Determine its capacitance per unit length.

Question 56

If the space between the inner and outer conductors of the coaxial structure in problem P.3-29 is free space, find the expression for V ( r ) in the region a r b by solving Laplace's equation. From V ( r ) , obtain the surface charge densities on the conductors and the capacitance per unit length of the structure. Compare your result with Eq. (3-90).

Question 57

The cathode-ray oscilloscope (CRO) shown in Fig. 3-2 is used to measure the voltage applied to the parallel deflection plates.
a) Assuming no breakdown in insulation, what is the maximum voltage that can be measured if the distance of separation between the plates is h
b) What is the restriction on L if the diameter of the screen is D
c) What can be done with a fixed geometry to double the CRO's maximum measurable voltage
Fig Electrostatic deflection system of a cathode-ray oscilloscope (Example 3-2)
Example

Question 58

State Coulomb's law.

Question 59

Assume that two homogeneous isotropic dielectric media with dielectric constants r l = 3 and r 2 = 2 are separated by the xy -plane. At a common point, E 1 = a x a y 5 a z 4. Find E 2 , D 2 , 1 , and 2.

Question 60

Solve the following problems:
a) Find the breakdown voltage of a parallel-plate capacitor, assuming that the conducting plates are 50 (mm) apart and the medium between them is air.
b) Find the breakdown voltage if the entire space between the conducting plates is filled with plexiglass, which has a dielectric constant 3 and a dielectric strength 20 (kV/mm).
c) If a 10-(mm) thick plexiglass is inserted between the plates, what is the maximum voltage that can be applied to the plates without a breakdown

Question 61

Define capacitance and capacitor.

Question 62

A fixed voltage is connected across a parallel-plate capacitor.
a) Does the electric field intensity in the space between the plates depend on the permittivity of the medium
b) Does the electric flux density depend on the permittivity of the medium Explain.

Question 63

Write the differential form of the fundamental postulates of electrostatics in free space.

Question 64

An electric dipole at the origin has a dipole moment a z 0.1 (nC · m). Find V and E at (a) (0, 0, 5(m)), and (b) (2(m), /3, /8).

Question 65

A finite line charge of length L carrying uniform line charge density t is coincident with the x -axis.
a) Determine V in the plane bisecting the line charge.
b) Determine E from directly by applying Coulomb's law.
c) Check the answer in part (b) with V

Question 66

What is the difference between the dielectric constant and the dielectric strength of a dielectric material

Question 67

A spherical capacitor consists of an inner conducting sphere of radius R i and an outer conductor with a spherical inner wall of radius R o. The space in between is filled with a dielectric of permittivity . Determine the capacitance.

Question 68

An infinite conducting cone of half-angle a is maintained at potential V 0 and insulated from a grounded conducting plane, as illustrated in Fig. 3-33. Determine
a) the potential distribution V ( ) in the region / 2,
b) the electric field intensity in the region / 2, and
c) the charge densities on the cone surface and on the grounded plane.

Question 69

Given two point charges: q 1 = 10( C) at (2, 0, 4) and q 2 = 60( C ) at (0, 1, 2), determine
a) the electric field intensity at q 1 due to q 2 , and
b) the magnitude of the force experienced by q 1 All dimensions are in meters.

Question 70

A line charge of uniform density forms a semicircle of radius b in the upper half xy -plane. Determine the magnitude and direction of the electric field intensity at the center of the semicircle.

Question 71

Define polarization vector. What is its SI unit

Question 72

Show that the following potential functions satisfy two-dimensional Laplace's equation:
a) Ae kx sin ky, and
b)

where A, k, a, and b are constants.

Question 73

Write the capacitance formula for a parallel-plate capacitor of area S whose plates are separated by a medium of dielectric constant r and thickness d.

Question 74

Assume that fixed charges + Q and Q are deposited on the plates of an isolated parallel-plate capacitor.
a) Does the electric field intensity in the space between the plates depend on the permittivity of the medium
b) Does the electric flux density depend on the permittivity of the medium Explain.

Question 75

Three 2-( C) point charges are located in air at the corners of an equilateral triangle that is 10 (cm) on each side. Find the magnitude and direction of the force experienced by each charge.

Question 76

State Gauss's law. Under what conditions is Gauss's law especially useful in determining the electric field intensity of a charge distribution

Question 77

Determine the capacitance of the parallel-plate capacitor in Fig. 3-20 by starting with an assumed potential difference V 12 between the upper and lower plates, then finding Q and taking the ratio Q/V 12.
Fig A unit cube (Example 2-12)

Example

Question 78

Assume that the z = 0 plane separates two lossless dielectric regions with r l = 2 and r 2 = 3. If we know that E 1 in region 1 is a x 2 y a y 3 x + a z (5 + z ), what do we also know about E 2 and D 2 in region 2 Can we determine E 2 and D 2 at any point in region 2 Explain.

Question 79

Three capacitors 1 ( F), 2 ( F), and 3 ( F)are connected as shown in Fig. 3-31 across a 120-volt source. Calculate the electric energy stored in each capacitor.

Question 80

For a positive point charge Q located at a distance d from each of two grounded perpendicular conducting half-planes shown in Fig. 3-34, find the expressions for
a) the potential and the electric field intensity at an arbitrary point P ( x, y ). and
b) the surface charge densities induced on the two half-planes.