   Chapter 16.2, Problem 42E

Chapter
Section
Textbook Problem

The force exerted by an electric charge at the origin on a charged particle at a point (x, y, z) with position vector r = ⟨x, y, z⟩ is F(r) = Kr/| r |3 where K is a constant. (See Example 16.1.5.) Find the work done as the particle moves along a straight line from (2, 0, 0) to (2, 1, 5).

To determine

To find: The work done by the particle when it moves along a straight line from the point (2,0,0) to (2,1,5).

Explanation

Given:

The force field is F(r)=Kr|r|3.

The position vector is r=x,y,z.

The particle moves along a straight line from the point (2,0,0) to (2,1,5).

Formula used:

The work done by the force field F(x,y,z) along the line segment.

CFdr=abF(r)rdt (1)

Where, r is the position vector of the particle, r is the derivative of position vector, a is the lower limit of scalar a parameter, and b is the upper limit of the scalar parameter.

The parametric equations for a line segment through the point (x0,y0,z0) and parallel to the direction vector v=a,b,c.

x=x0+at,y=y0+bt,z=z0+ct (2)

The director vector v=a,b,c for a line segment from the point (x0,y0,z0) to (x1,y1,z1).

a,b,c=x1x0,y1y0,z1z0 (3)

To calculate the vector v=a,b,c substitute 2 for x0, 0 for y0, 0 for z0, 2 for x1, 1 for y1, and 5 for z1 in equation (3),

a,b,c=22,10,50=0,1,5

Substitute 2 for x0, 0 for y0, 0 for z0, 0 for a, 1 for b, and 5 for c in equation (2),

x=2+(0)t,y=0+(1)t,z=0+(5)tx=2,y=t,z=5t

Consider the limits of scalar parameter t are from 0 to 1 that is 0t1.

Write the position vector from the parametric equations as follows.

r=2,t,5t

Write the equation of force field as follows.

F(r)=Kr|r|3 (4)

Substitute the position vector 2,t,5t for r in equation (4),

F(r)=K2,t,5t|2,t,5t|3=K2,t,5t|22+t2+(5t)2|3=K(4+26t2)322,t,5t

To find the derivative of the vector function, differentiate each component of the vector function.

Differentiate each component of the vector function r(t)=2,t,5t as follows

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