Electric Fields
We learned that charged particles can attract and repel one another and all at a distance! Another way of viewing this situation is to say that a charged particle creates a condition around it which can be senced by any other charged particles that come into the proximity of it. We quantify this condition by the electric field. The electric field is simply given by the ratio of force to charge when a test particle, q, that does not perturb the charge distribution is introduced: E= F q
Units for electric field are Newton per Coulomb. Electric field from a point charge From the definition we have E= 1 q ˆ r 4πε 0 r 2
The electric field points away from a positively charged particle. It points in the direction of a negatively charged particle. The superposition principle holds for the total field from a collection of many point charges that are indexed by i: E = ∑ Ei
i
Hair pointing along the electric field. We demonstrate the electric field with the Van de Graaff generator. Charge me up and hair stands on end pointing in the direction of the electric field. As the body charges an electric field builds up around me. The electric field polarizes the molecules in my hair and these in turn line up in the electric field. They also show the direction of the electric field vector. The magnitude of the electric field controls how many hairs stand on end. How many hairs on end per unit area is a measure of the strength of the electric field.
Collin Leslie Broholm
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Sparking or Electrical Breakdown When the electric field in air reaches approximately 3 × 10 6 N/C it is strong enough to accelerate any stray electron to the point that it can ionize air at ambient pressure. Further charge is again accelerated and the result is an ark or a spark. A stream of charged particles that moves along the field lines. Lightning produced by the van der Graaff generator. We use the generator to separate charge on two spheres. As the spheres approach each other a force of attraction is felt. If they get very close the electric field produced between the spheres exceeds the breakdown field and a discharge allows charge to flow between the two spheres to neutralize both. Electric field from a dipole Just before the spark flies the two oppositely charged spheres represent an electric dipole. When we hung a balloon on the wall, I argued that molecules and atoms in the wall were being polarized by a slight shift of the negative and positive charge with respect to one another. These again represent an electric dipole. In genereal an electric dipole is a neutral object in which the center of the positive and negative charge distribution are shifted with respect to each other by an amount that we typically denote d. The electric dipole is not just how we hang a party balloon the next simplest thing up from a point charge. The dipole is important for understanding how all these things work: • The dipole antenna • How many organic solids hold themselves together (van der Waals crystal) • The behavior of complex molecular fluids including polymers. Electric field visualization There is an interesting electric field distribution around an electric dipole. To view it we use a Wimshurst machine to separate positive from negative charge and form a dipole (explain). We use oil with fibers that are easily to directly view the electric dipole field. Quantitative Expression for the Dipole Field. We use the superposition theorem to write the expression for the electric dipole field on the axis of displacement (see book for derivation and figure 23.8) E(z ) = ∑ E i ≈
i
1 p 2πε 0 z 3
In this expression z is the distance from the place where we are measuring the electric field to the center of the dipole. The approximation is good when z >> d which is important for most of the interesting applications. p = Qd is the dipole moment. It is a vector quantity that point from the negative to the positive charge in the dipole. We have only written the expression for the electric field along the dipole axis. Along this axis the dipole field far from the dipole always points along the same p direction. This is different Collin Leslie Broholm Page 2 01/30/00
from a point charge where the direction of the field reverses on opposite sides of the charge. A positive charge on the dipole axis experiences a force in the direction of the dipole, a negative charge experiences a force opposite to this direction.
We notice that the dipole field decreases with the third power of distance rather than the second power as for the field from a point charge. The field decreases more rapidly as we move away from a dipole than when we move away from a point charged object. Effect of Uniform electric field on Dipole moment Turn things around and consider what happens when we subject a dipole to a uniform field. The force on the dipole is obtained from superposition: F = ∑ Fi = E(Q − Q ) = 0
i
no net force so a uniform field cannot cause translation of a dipole. However the field exerts a torque which we calculate with respect to the center of the dipole as our point of reference: ! (− d ) × E(− Q ) = p × E d τ = ∑ ri × Fi = × EQ + i 2 2 Torque only vanishes when the dipole moment is parallel to the electric field. The stable equilibrium position corresponds to p || E . We reconsile this result with • The party balloon experiment • The electric field visualization device based on induced dipole moment in fibers • hair standing on end on charged individual.
Electric field from continuous charge distribution When we have lots of point charges distributed in some manner it is inefficient and impossible to specify the exact location of every charged particle. Instead we might state the charge density per unit length on a wire or the charge density per unit area on a plane. Using superposition the expression for the electric field becomes an integral rather than a summation. Specifically for volume surface and line charge distributions we can write the following expressions for the electric field based simply on the principle of superposition: Collin Leslie Broholm Page 3 01/30/00
E(r ) =
ρ (r ')d 3 r " 1 R ∫ 4πε 0 V ' R 2 2 " (r ) = 1 ∫ σ (r ')2d r R E 4πε 0 S ' R λ (r ')dr " 1 E(r ) = R ∫ 4πε 0 S ' R 2
for volume charge distribution
for surface charge distribution
for line charge distribution
In all these expressions R is the vector pointing from the active point in the integration ˆ ( r ' ) to the point r where we are evaluating the electric field: R = r − r ' . R is the corresponding unit vector. To illustrate the use of these expressions we look at two examples: Field from a charged segment of wire Charge is distributed uniformly with charge density per length unit λ (units C/m). We calculate the field at a distance r from the center of the wire in the perpendicular direction. At this location because of the symmetry of the problem the electric field is perpendicular to the wire. The components of field along the wire from opposite ends of the wire cancel. Thus we only add up the components of E that are perpendicular to the wire:
r2 + z2
z r
2 λ dz 1 cosθ E (r ) = ∫ 2 4πε 0 − L 2 r + z 2 L
ˆ Here cosθ comes from the projection of R on the direction perpendicular to the wire. This can be written in terms of r and z as follows:
Collin Leslie Broholm
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E (r ) =
2 λ rdz ∫ 4πε 0 − L 2 (r 2 + z 2 )3 2 L L
2r λ du = ∫ 4πε 0 r − L 2 r ( + u 2 )3 2 1 2 λ u r = 4πε 0 r 1 + u 2 − L 2r L
=
Q 4πε 0 r 2
1+ L
( 2r )
1
2
Where Q=λL is the total charge on the wire. We notice that the expression makes sense in the two extreeme limits. Very far from a short wire the field generated looks like that for a point charge because L → 0 and so the last fraction approaches unity. Very close 2r to a long wire we have 1 + L
( 2r )
2
→L
2r
and this leads to
E (r ) =
λ 2πε 0 r
as the electric field close to a long wire with charge λ per unit length.
Collin Leslie Broholm
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