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Chapter 26, Problem 24PQ

Two point charges, q1 = −2.0 μC and q2 = 2.0 μC, are placed on the x axis at x = 1.0 m and x = −1.0 m, respectively (Fig. P26.24).

  1. a. What are the electric potentials at the points P (0, 1.0 m) and R (2.0 m, 0)?
  2. b. Find the work done in moving a 1.0-μC charge from P to R along a straight line joining the two points.
  3. c. Is there any path along which the work done in moving the charge from P to R is less than the value from part (b)? Explain.

(a)

Expert Solution
Check Mark
To determine

The electric potential at (0,1.0m), and (2.0m,0).

Answer to Problem 24PQ

The electric potential at P is 0_, and electric potential at R is 1.2×102V_.

Explanation of Solution

Write the expression for electric potential due to two charges.

  V=kq1r1+kq2r2                                                                                                             (I)

Write the expression for distance between two points.

  r=(x2x1)2+(y2y1)2                                                                                      (II)

Conclusion:

Consider the figure 1 below. r1 is the distance from q1 to P, and r2 is the distance from q2 to P

Webassign Printed Access Card For Katz's Physics For Scientists And Engineers: Foundations And Connections, 1st Edition, Single-term, Chapter 26, Problem 24PQ , additional homework tip  1

Substitute, 1.0m for x2, 0 for x1, 0 for y1, and 1.0m for y2 in equation (II) to obtain distance r1.

  r1=(1.0m0)2+(01.0m)2=(1m)2+(1m)2=2

Substitute, 1.0m for x2, 0 for x1, 0 for y1, and 1.0m for y2 in equation (II) to obtain distance r2.

  r2=(1.0m0)2+(01.0m)2=(1m)2+(1m)2=2

Substitute, 2m for r1, and r2, 2.0×106C for q1, and 2.0×106C for q2 in equation (I) to obtain the electric potential at P.

  VP=k(2.0×106C2+2.0×106C2)=0

Consider figure 2 given below. r1 is the distance from q1 to R, and r2 is the distance from q2 to R

Webassign Printed Access Card For Katz's Physics For Scientists And Engineers: Foundations And Connections, 1st Edition, Single-term, Chapter 26, Problem 24PQ , additional homework tip  2

Substitute, 2.0m for x2, 1.0m for x1, 0 for y1, and 0 for y2 in equation (II) to obtain distance r1.

  r1=(2.0m1.0m)2+(00)2=(1m)2=1m

Substitute, 2.0m for x2, 1.0m for x1, 0 for y1, and 0 for y2 in equation (II) to obtain distance r2.

  r2=(2.0m(1.0m))2+(00)2=(9m)2=3m

Substitute, 8.99×109Nm2/C2 for k, 1m for r1, 3.0m for r2, 2.0×106C for q1, and 2.0×106C for q2 in equation (I) to obtain the electric potential at R.

  VR=8.99×109Nm2/C2(2.0×106C1m+2.0×106C3.0m)=1.2×104V

Therefore, the electric potential at P is 0_, and electric potential at R is 1.2×104V_.

(b)

Expert Solution
Check Mark
To determine

The work done in moving a 1.0μC charge from P to R along a straight line joining the two points.

Answer to Problem 24PQ

The work done in moving a 1.0μC charge from P to R along a straight line joining the two points is 1.2×102J_.

Explanation of Solution

The work done will be equal to change in potential energy between two points.

Write the expression for change in electric potential energy.

  WPR=ΔUE=q(VRVP)                                                                                                     (I)

Here, q is the charge of the particle.

Conclusion:

Substitute, 1.0×106C for q, 1.2×104V for VR, and 0 for VP in equation (I).

  WpR=1.0×106C(1.2×104V0)=1.2×102J

Therefore, the work done in moving a 1.0μC charge from P to R along a straight line joining the two points is 1.2×102J_.

(c)

Expert Solution
Check Mark
To determine

The path along which the work done is less than the value obtained in part (b).

Answer to Problem 24PQ

No, there is no other path through which the charge can move so that the work done is less than the value obtained part (b).

Explanation of Solution

Write the expression for work done in terms of change in potential energy.

  W=ΔUE=qΔV

The work done is depends on charge and the change in potential, and does not depends on the path followed by the particle.

Hence, there is no other path in which work done is less than 1.2×102J.

Conclusion:

Therefore, the work done is not depends on the path followed by the particle, and only depends on the change in potential. There is no other path with work done less than 1.2×102J.

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Chapter 26 Solutions

Webassign Printed Access Card For Katz's Physics For Scientists And Engineers: Foundations And Connections, 1st Edition, Single-term

Ch. 26 - Try to complete Table P26.4 from memory. If you...Ch. 26 - Try to complete Table P26.5 from memory. If you...Ch. 26 - Can you associate electric potential energy with...Ch. 26 - Consider the final arrangement of charged...Ch. 26 - Using the usual convention that the electric...Ch. 26 - FIGURE P26.8 A Find an expression for the electric...Ch. 26 - A hydrogen atom consists of an electron and a...Ch. 26 - What is the work that a generator must do to move...Ch. 26 - How far should a +3.0-C charged panicle be from a...Ch. 26 - A proton is fired from very far away directly at a...Ch. 26 - Four charged particles are at rest at the corners...Ch. 26 - FIGURE P26.14 Problems 14, 15, and 16. Four...Ch. 26 - Four charged particles are at rest at the corners...Ch. 26 - Eight identical charged particles with q = 1.00 nC...Ch. 26 - A conducting sphere with a radius of 0.25 m has a...Ch. 26 - The speed of an electron moving along the y axis...Ch. 26 - Figure P26.20 is a topographic map. a. Rank A, B,...Ch. 26 - At a point in space, the electric potential due to...Ch. 26 - Explain the difference between UE(r) = kQq/r and...Ch. 26 - Suppose a single electron moves through an...Ch. 26 - Two point charges, q1 = 2.0 C and q2 = 2.0 C, are...Ch. 26 - Separating the electron from the proton in a...Ch. 26 - Can a contour map help you visualize the electric...Ch. 26 - Prob. 27PQCh. 26 - Find the electric potential at the origin given...Ch. 26 - Prob. 29PQCh. 26 - Prob. 30PQCh. 26 - Prob. 31PQCh. 26 - Prob. 32PQCh. 26 - A source consists of three charged particles...Ch. 26 - Two identical metal balls of radii 2.50 cm are at...Ch. 26 - Figure P26.35 shows four particles with identical...Ch. 26 - Two charged particles with qA = 9.75 C and qB =...Ch. 26 - Two charged particles with q1 = 5.00 C and q2 =...Ch. 26 - Prob. 38PQCh. 26 - Prob. 39PQCh. 26 - A uniformly charged ring with total charge q =...Ch. 26 - A line of charge with uniform charge density lies...Ch. 26 - A line of charge with uniform charge density =...Ch. 26 - A Consider a thin rod of total charge Q and length...Ch. 26 - Figure P26.44 shows a rod of length = 1.00 m...Ch. 26 - The charge density on a disk of radius R = 12.0 cm...Ch. 26 - Prob. 46PQCh. 26 - In some region of space, the electric field is...Ch. 26 - A particle with charge 1.60 1019 C enters midway...Ch. 26 - Prob. 49PQCh. 26 - Prob. 50PQCh. 26 - Prob. 51PQCh. 26 - Prob. 52PQCh. 26 - Prob. 53PQCh. 26 - According to Problem 43, the electric potential at...Ch. 26 - The electric potential is given by V = 4x2z + 2xy2...Ch. 26 - The electric potential V(x, y, z) in a region of...Ch. 26 - Prob. 57PQCh. 26 - In three regions of space, the electric potential...Ch. 26 - Prob. 59PQCh. 26 - Prob. 60PQCh. 26 - The distance between two small charged spheres...Ch. 26 - Prob. 62PQCh. 26 - A glass sphere with radius 4.00 mm, mass 85.0 g,...Ch. 26 - Prob. 64PQCh. 26 - Two 5.00-nC charged particles are in a uniform...Ch. 26 - A 5.00-nC charged particle is at point B in a...Ch. 26 - A charged particle is moved in a uniform electric...Ch. 26 - Figure P26.68 shows three small spheres with...Ch. 26 - What is the work required to charge a spherical...Ch. 26 - For a system consisting of two identical...Ch. 26 - Figure P26.71 shows three charged particles...Ch. 26 - Problems 72, 73, and 74 are grouped. 72. A Figure...Ch. 26 - A Start with V=2k[(R2+x2)x] for the electric...Ch. 26 - A Review Consider the charged disks in Problem 72...Ch. 26 - A long thin wire is used in laser printers to...Ch. 26 - An electric potential exists in a region of space...Ch. 26 - A disk with a nonuniform charge density =ar2 has...Ch. 26 - An infinite number of charges with q = 2.0 C are...Ch. 26 - An infinite number of charges with |q| =2.0 C are...Ch. 26 - Figure P26.80 shows a wire with uniform charge per...Ch. 26 - Prob. 81PQ
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