Lectures 7 & 8

Electric Potential – continuous charge distributions

•To calculate E produced by an electric dipole•To investigate the forces and torques an electric dipole experiences in an external E-field

Importance:

•Production and reception of radio and TV signals

• Interaction of molecules with EM radiation:

- molecular spectroscopy

- trace analysis

Electric Dipoles

V of an Infinite Wire of Uniform Charge per Unit Length

rE

02πε

=

V of a spherical volume of uniform charge, of radius a, carrying a total charge Q

Rr ≤(i)

Rr ≥(ii)

Q

R

l

+q -q

•V due to the two charges at P

−+

−=r

q

r

qV

00P 44 πεπε

Assumption: r >> lθcos

2

lrr ±≈±

r-r+

P

θr

l/2

Electric Dipole: Calculation of the E-field at an arbitrary (r, θ)

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−=

θ

θπε 2

220

P

cos4

cos

4 lr

lqV

204 r

cosp

πεθ

≈

r

VEr ∂

∂−= P

304

cos2

r

p

πεθ

=θ∂

∂−=

P1 V

rET 3

04

sin

r

p

πεθ

=

-q

+q

pl

E qE

-qE

No Net Force

But Torque - rotates the dipole clockwise

An Electric Dipole in an External E-Field

Fr∧=

Torque (of a couple)

The resultant torque is:

( ) FdxFdxF =×−+×

Torque of a couple is the same about any axis drawn perpendicular to the plane it defines

The magnitude of the torque of a couple is calculated from

Fd=

-q

+q

l

qE

-qEθ

θθ

sinpE

sinqlEsinlqEdqE

=

=×=×=

θ

d

The torque tends to align p and E

Ep∧=

+q

qE

-qE -q

The P.E. of an electric dipole in an E-field

work done change in P.E

Work done by during an infinitesimal displacement dθ:

θddW =The torque is in the direction of decreasing θ θ sinpE−=

θθθ dsinpEddW −==Hence

Finite displacement from θ1 to θ2:

( ) θθθ

θ

dsinpEdWW 2

1

∫ ∫ −==

Therefore12 θθ cospEcospEW −=

Work = -change of P.E.21 UUW −=

Thus the P.E. of an electric dipole in an E-field is:

E.pcospEU −=−= θ

Minimum at θ = 0, maximum at θ = π, and zero at θ = π/2

Review and Summary

•An electric dipole is a pair of electric charges of equal magnitude q but opposite sign, separated by a distance l

•The electric dipole moment is defined to have magnitude p = ql

•We calculate the E of an electric dipole at any position in space by a method far easier than using Coulomb’s law and superposition