Tipler: 21-7

Electromagnetism I

Electric Dipoles and their Interactions in

Electric Fields

An electric dipole consists of a positive charge separated from a negative charge of the same magnitude by a small distance. Which, if any, of the diagrams best represents the electric field lines around an electric dipole?

ET

EP

Learning Objectives

•To calculate E produced by an electric dipole

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

An electric dipole:

l

+q -q

Electrically neutral

A Reminder of Electric Dipoles

Define dipole moment as p = ql

The vector of p is drawn from the negative to the positive point charge

p is a vector.

p

Molecular example: H2O

p = 6.1 x 10-30 C m •The electric dipole of water makes it an excellent solvent

•Used in heating food in a microwave oven

Calculation of the E-field at an arbitrary (r, )

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎠⎞

⎜⎝⎛ +

−

⎟⎠⎞

⎜⎝⎛ −

= 220

22

4

1

ar

q

ar

qE

πε

The E-field due to an Electric Dipole - Calculation

To simplify the calculation, we will only compute the field along the axis

r

E

-q +q

a

a/2

304

2

r

qaE

πε≈

For r >> a

This applies to any points along the line of the dipole,

and only for points along the line of the dipole.

Along this particular direction, the E field from the

positive charge is in opposite direction to that from

the negative charge.

To obtain E-field:

1) Coulomb’s Law, involving vector sums.

2) Gauss’s law, if the charge distribution

has a high degree of symmetry.

3) Get V first, then differentiate. No vector sums.

How to make a decision:

Choose the method that consumes the

least number of brain cells.

l

+q -q

•V due to the two charges at P

−+

−=r

q

r

qV

00P 44 πεπε

Assumption: r >> l

cos2

lrr ±≈±

r-r+

P

r

l/2

No nasty vectors here.

€

VP =q

4πε0

−lcosθ

r2 −l2

4cos2 θ

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

Hence

€

≈−pcosθ

4πε0r2

as r >> l

The sign of Vp depends on .

⎟⎠

⎞⎜⎝

⎛∂∂

+∂∂

−=

ˆV

rr̂

r

VE

1

In Plane Polar coordinates (lecture 4)

€

≈−pcosθ

4πε0r2PV

r

VEr ∂

∂−= P

∂∂

−=P1 V

rET€

=−2pcosθ

4πε0r3

€

=−psinθ

4πε0r3

l

+q l/2

r-r+

P

r

-q

Er points towards the centre of the dipole

ET points towards decreasing .

Er change signwhen is greater than 90 degrees.

€

=−2pcosθ

4πε0r3

€

=−psinθ

4πε0r3

€

Er =−2pcosθ

4πε0r3

€

ET =−psinθ

4πε0r3

Electric Dipoles in Uniform Electric Fields

(Tipler 671-672)

-q

+q

pl

E qE

-qE

No Net Force

But Torque - rotates the dipole clockwise

Two equal and opposite forces whose lines of action do not coincide constitute a couple. The two forces always

have a turning effect, called a torque

€

=r × F

A torque is defined as the moment of a force

Mathematically (STMR)

Torque (of a couple) (Tipler – pages 309-311)

The resultant torque is:

( ) FdxFdxF =×−+×

The magnitude of the torque of a couple is calculated from

Fd=i.e. torque = one force perpendicular distance between forces

-q

+q

l

qE

-qE

Torque

sinpE

sinqlEsinlqEdqE

=

=×=×=

d

The torque tends to align p and E

In vector form:

€

r = rp ×

r E = pE sinθ

The direction of

p

E

An electric dipole of moment p is placed in a uniform external electric field. The dipole moment vector is in the positive y direction. The external electric field vector is in the positive x direction. When the dipole is aligned as shown in the diagram, the net torque is in the

A)positive x direction. B)positive y direction. C)negative x direction.   D)positive z direction. E)negative z direction.

Electric potential energy of a dipole

+q

-q

E

Electric potential at +q is V+, the potential energy is qV+

Electric potential at -q is V-, the potential energy is -qV-

The total potential energy of the dipole is:

€

U = qV+ − qV− = q(V+ −V−) = q(E .d) = qlcosθ • E

= pE cosθ

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

E.pcospEU −=−=

Minimum at = 0, maximum at = π,

and zero at = π/2

The electric dipole is like an electric version of

a compass

P

E

The potential energy of a magnetic dipole is

€

U = −r μ •

r B

What is the torque on the dipole for

the above configuration?

€

= pE sinθ

Cooking instructions:

Molecules with dipole moment,

Molecules that are mobile

H2O= 6.1 x 10-30 C m

Electric Dipole Moment of

Some Gas Molecules

HCl 1.1

HBr 0.8

H2O 1.8

SO2 1.6

N2O 0.2

NH3 1.5

•An electric dipole in an electric field experiences a torque:

•The potential energy for an electric dipole in an electric field E depends on the orientation of the dipole moment p with respect to the field:

E.pU −=

Ep ∧=

p = 0.02 e.nm

E = 3 × 103 N/C

Calculate

(a) the magnitude of the torque

(b) The potential energy

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