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Physics 108

Review - Electric Fields


(a) Find the magnitude and direction of the force of +Q on qo at (i) P1 and (ii) P2 in Fig 1a below. (b) Find the magnitude and direction of the force of -Q on qo at (i) P1 and (ii) P2 in Fig.1b below. Take Q = 2 x 10-6 C and qo = 10-12 C.


Now imagine that all the +qo's are removed from Fig. 1 above.  Find the magnitude and the direction of the electric field (a) in Fig. 1a and (b) in Fig. 1b at P1 and P2.


(a) Find the electric field at point P in Fig. 2 below. (b) Repeat for q2 = +1 nC.


In Fig. 3 below,  q1 = +1.00 µC and q2 = -0.0800 µC.  Find the direction and magnitude of the electric field at point P.


Find (a) the electric field, direction and magnitude, at P (4.00 m, 5.00m) in Fig. 4 below due to q1 = +5.00 x 10-6 C at (0, 2.00 m), q2 = -3.00 x 10-6 C at
(4.00 m, 0), and q3 = +1.6 x 10-6 C at (0, 5.00 m) and (b) the force on q4 =
+2.0 x 10-6 C placed at P.


A straight positively charged line coincides with the X-axis and carries a charge per unit length of l. While the charge is actually discrete, it is useful to think of the charge spread continuously along the line.


(a) the charge dq on a length dx of the line,

(b) the magnitude of dE1 or dE2 at point P along the Y-axis for the symmetrically displaced elements of length dx1 and dx2 in Fig. 5 below,

(c) the direction of dE1 + dE2,  and

(d) the magnitude of the component of dE1 or dE2 in the direction of the resultant field.

Now we can integrate to find the resultant field. Remember that integration is not a vector addition, so we cannot integrate until we have found the resultant direction of the field.

(e) In Fig. 5 we have labeled various lengths and an angle: b, x, r, and Q. Which of these is a constant?

(f) Write x in terms of b and a function of Q.

(g) Find an expression for the resultant dEy in terms of k, l, b, a function of Q and dQ.

(h) If you were integrating with respect to x, the lower limit would be -3b/4 and the upper limit would be +3b/4. What are the lower and upper limits on the sine of Q?

(i) Complete the calculation of the resultant electric field at P in Fig. 5 below.


A thin nonconducting wire is bent into a semicircle of radius R  (Fig. 6 below). The wire is charged so that the linear density  l = lo cos Q,  where lo is a constant.  Find the direction and the magnitude of the electric field at the center of the semicircle at point O.


A cube with sides of 0.20 m has its center at the origin of a rectangular coordinate system and its faces perpendicular to the coordinate axes. The electric field is E = 100 N/C-m xi.  Determine the amount of charge within the cube.


In Fig. 7 below, the dashed spherical surface is a Gaussian surface of radius r. The solid sphere is a uniform distribution of charge of radius R. If spherical symmetry is to be preserved, the electric field E at P remains constant when the solid sphere is rotated. If this is true, (a) can the electric field depend on angles Q or F? (b) must the electric field E always be radially outward (or inward if the charge is negative)? and (c) must the electric field be constant in magnitude everywhere on the Gaussian surface of radius r? (d) If the angle between the electric field and the surface,   E, dA,  is zero how can you write E . dA? Remember the direction of the surface is always taken as the outward normal. (e) If E is a constant, how may you write for the spherical Gaussian surface?


The thick, spherical shell of inner radius a and outer radius b shown in Fig. 8 below carries a uniform volume charge density r with total charge Q.


(a) r in terms of Q, a, and b,

(b) the charge enclosed by a spherical Gaussian surface of radius (i)  r < a,
     (ii)  a < r < b,  and  (iii)  r > b,  and

(c) the electric field for (i)  r < a,  (ii)  a < r < b,  and (iii)  r > b.

(d) Use the expressions found in (c) to show that these also are the electric      field for  (i)  r ≤ a,  (ii)  a ≤ r ≤ b,  and  (iii)  r ≥ b.


What is (a) the lateral surface area and (b) the volume of a right circular cylinder of radius r and length L?


Use Gauss's theorem to determine the electric field due to a very long wire carrying a charge per unit length +l a distance r from the wire.


The dashed spherical surface in Fig. 9 is a Gaussian surface, which surrounds a dipole. The solid curves with arrows represent the field lines for the dipole. Can Gauss's theorem be used to find the field of a dipole? Explain your answer.


The electric field outside and at a distance of 0.5 m from the center of a charged sphere, the axis of a long charged cylinder or an infinite sheet of charge is 144 N/C.  For which of these charge distributions do you find an electric field of (a) 72 N/C, (b) 36 N/C, (c) 144 N/C at a distance of 1.0 m?  Explain your answers. Find (d) the charge q on the sphere, (e) the charge per unit length l on the cylinder and (f) the charge per unit area s on the infinite sheet.


Three infinite sheets of charge are parallel to each other, as is shown in Fig. 10. The sheet on the left has a uniform surface charge density +s,  the one in the middle a uniform surface charge density - s , and the one on the right a uniform surface charge density of +s.  Find the electric field at  (a) P1,  (b) P2,  (c) P3, and (d) P4.


A conducting sphere of charge +Q and radius a is concentric with a very thin conducting spherical shell of charge - Q and radius  b > a.  (a) Where are the charges?  Find the electric field for  (b)  r ≤ a,  (c)  a ≤ r ≤ b,  (d)  r ≥ b.


An infinitely long insulating cylinder of radius R has a volume charge density that varies with the radius as r = ro(a - cr), where ro,  a,  and c are positive constants and r is the distance from the axis of the cylinder.  Find the electric field at radial distances for  (a)  r < R  and (b)  r > R.


Can electric field lines intersect?  Why or why not?


A solid sheet of nonconducting material of thickness t and infinite length and width has a uniform positive charge density r throughout the sheet. (a) Use Gauss's law to find the electric field as a function of the distance x from the center of the sheet for - t/2 < x < t/2.  (b) If a small hole is drilled though the sheet and a particle of mass m and charge -q is released from rest on one side of the sheet, what is the frequency of the particle's motion?


A uniform electric field E is set up between two metal plates of length L and spacing d, as shown in the figure to the right. An electron enters the region midway between the plates moving horizontally with speed v.  Find an expression for the minimum speed the electron must have to get through the region without hitting either plate.  Neglect gravitational effects.


An electron is moving in a circular path around a long, uniformly charged wire carrying 2.5 nC/m. What is the speed of the electron?


A small sphere with mass m carries charge q. It hangs from a silk thread that makes an angle Q with a large, charged nonconducting sheet, as illustrated in the figure below. Find the sheet's surface charge density s.


Two identical dipoles are placed in a straight line as shown in Fig. 11a below. Find the direction of the electric force on each dipole in Fig. 11a.  Repeat for
Fig. 11b below.


An electric dipole consisting of charges +3.2 x 10-19 C and -3.2 x 10-19 C separated by 2.0 x 10-9 m is in a field of 5.0 x 105 N/C. Calculate the torque on the dipole when the dipole moment is (a) parallel and in the same direction as the field (b) perpendicular to the field and (c) parallel and in the opposite direction of the field.


Two positive point charges repel each other. Explain this experimental result in terms of the electric field of one charge action on the other charge.


There are really two types of problems concerning electric fields:
  1. Given a distribution of charges, find the electric field due to them.

  2. Given an electric field E, find the electric force Fe on a charge q.
Explain two ways of calculating electric fields and how you find the electric force on a charge q in an electric field.


One of two identical nonconducting rods of length L carries a total positive charge Q and the other carries a total charge of - Q as shown in Fig. 12 below. (a) Find the electric field at point P in Fig. 12 due to these charge distributions as a function of x. (b) What does this expression in (b) reduce to for x >> L. (c) Identify a dipole moment for this distribution.

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Susan D. Kunk
Phyllis J. Fleming
August 8, 2002
February 19, 2003