Young, H. D., Freedman, R. A., & Ford, A. L. (2020). University Physics with Modern Physics (Most Editions). Pearson. Online Purchase [https://amzn.to/4f6yCuY]
[21.7] Electric Dipoles Point charges q_1=-4.5 nC and q_2=+4.5 nC are separated by 3.1 mm, forming an electric dipole.
[a] Find the electric dipole moment (magnitude and direction).
[b] The charges are in a uniform electric field whose direction makes an angle of 36.9° with the line connecting the charges. What is the magnitude of this field if the torque exerted on the dipole has magnitude 7.2×10^(-9) N⋅m?
[a][Step 1] List all given data, unknown data, related formulae and convert units.
q_1=-4.5 nC=-4.5 ×10^(-9) C
q_2=+4.5 nC=+4.5×10^(-9) C
r=3.1 mm=3.1×10^(-3) m
Magnitude of electric dipole moment
p=qd
[a][Step 2]
Take Note*
An electric dipole consists of two equal and opposite charges separated by a distance. The direction of the electric dipole moment is conventionally defined as pointing from the negative charge to the positive charge.
The direction of the electric field and the force in electrostatics is conventionally defined to go from positive to negative charge for several reasons:
An electric dipole is a pair of point charges with equal magnitude and opposite sign
∴p=qd=(4.5×10^(-9) C)(3.1×10^(-3) m)=1.4×10^(-11) C⋅m.
[b][Step 1]
Magnitude of the net torque
ϕ=36.9°,τ=7.2×10^(-9) N⋅m
τ=(qE)(d sinϕ )
=pE sinϕ
[b][Step 2] Assemble and execute.
This is not a difficult question by any means but as always, it is the detail that can throw us off, details such as the angle. Do we use the provide angle or alter it?
In this case, we can look at the electric field as the +x-axis.
E=τ/(qd sinϕ )
=(7.2×10^(-9) N⋅m)/([(1.4×10^(-11) C⋅m) sin〖(180°-36.9°)] 〗 )≈856.543 N/C
∴E=8.6×10^2 N/C.