The figure (Figure 1) shows a thin rod of length L with total charge Q . A) Find

Question

The figure (Figure 1) shows a thin rod of length L with total charge Q.

A) Find an expression for the electric field strength on the axis of the rod at distance r from the center.

Express your answer in terms of the variables L, Q, r, and appropriate constants.

B)

Evaluate E at r = 3.1 cm if L = 5.0 cm and Q = 3.8 nC .

Express your answer to two significant figures and include the appropriate units.

Charge Q is uniformly distributed along a thin, flexible rod of length L. The rod is then bent into the semicircle shown in the figure (Figure 2) .

C) valuate the field strength if L = 11 cm and Q = 24 nC .

Express your answer with the appropriate units.

Explanation / Answer

part a:

consider a small segment dx on the linear charge at a distance of x from P.

x will vary from r-(L/2) to r+(L/2)

charge on this small segment=dq=(Q/L)*dx

electric field due to this charge dq=k*dq/x^2

where k=coloumb;s constant

dE=k*(Q/L)*dx/x^2

integrating to find total electric field,

E=-k*(Q/L)/x

using the limits from x=r-0.5*L to x=r+0.5*L

E=k*(Q/L)*((1/(r-0.5*L))-(1/(r+0.5*L)))

part b:
using the given values,

E=9*10^9*3.8*10^(-9)*(1/0.05)*((1/0.006)-(1/0.056))

=101785.714 N/C

part c:

linear charge density=Q/L

radius of the semicircle=r=L/(pi)

due to symmetry , the y component of electric field due to small segment on 2nd quadrant will cancel out small segment in 3rd quadrant

so total field will be along +ve x axis.

we can compute field for one quadrant only and then multiply by 2 to get total electric field strength.

consider a small segment of length r*d(theta)

where theta is the angle made with -ve x axis.

theta varies from 0 degree to 90 degree.

charge on this segment=dq=r*d(theta)*Q/L

=(Q/pi)*d(theta)

electric field at the center =k*dq/r^2

its component along x axis=k*dq*cos(theta)/r^2

=k*(Q/pi)*cos(theta)*d(theta)/r^2

integration of cos(theta)*d(theta)=sin(theta)

integrating with limits from theta=0 to theta=90 degrees

we get electric field due to 2nd quadrant =k*(Q/pi)/r^2

=k*Q/(pi*r^2)

=k*Q/(pi*L^2/pi^2)

=pi*k*Q/L^2

so total field due to the entire semicircle=2*pi*k*Q/L^2

using the values, we get electric field strength

=2*pi*9*10^9*24*10^(-9)/0.11^2=112162.646 N/C

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.