Fields

A field is a way of explaining action at a distance.  Massive particles attract each other.  How do they do this, if they are not in contact with each other?  We say that massive particles produce gravitational fields.  A field is a condition in space that can be probed.  Massive particle are also the probes that detect a gravitational field.  A gravitational field exerts a force on massive particles.  The magnitude of the gravitational field produced by a massive object at a point P is the gravitational force per unit mass it exerts on another massive object located at that point.  The direction of the gravitational field is the direction of that force

The magnitude of the gravitational force near the surface of Earth is F = mg, the gravitational field has magnitude F/m = g.  Its direction is downward.

Charged particles attract or repel each other, even when not in contact with each other.  We say that charged particles produce electric fields.  Charged particles are also the probes that detect an electric field.  An electric field exerts a force on charged particles.  The magnitude of the electric field E produced by a charged particle at a point P is the electric force per unit positive charge it exerts on another charged particle located at that point.  The direction of the electric field is the direction of that force on a positive charge.  The actual force on a particle with charge q is given by F = qE.  It points in the opposite direction of the electric field E for a negative charge.

In the presence of many other charges, a charge q is acted on by a net force F, which is the vector sum of the forces due to all the other charges.  The electric field due to all the other charges at the position of the charge q is E = F/q, i.e. it is the vector sum of the electric fields produce by all the other charges.  To measure the electric field E at a point P due to a collection of charges, we can bring a small positive charge q to the point P and measure the force on this test charge.  The test charge must be small, because it interacts with the other charges, and we want this interaction to be small.  We divide the force on the test charge by the magnitude of the test charge to obtain the field.

Consider a point charge Q located at the origin.
The force on a test charge q at position r is F = (k_eQq/r²) (r/r).
The electric field produced by Q is  E = F/q = (k_eQ/r²) (r/r).

If Q is positive, then the electric field points radially away from the charge.
The electric field decreases with distance as 1/(distance)².

If Q is negative, then the electric field points radially towards the charge.
The electric field decreases with distance as 1/(distance)².

We obtain the electric field due to a collection of charges using the principle of superposition.
E = E(Q₁) + E(Q₂) + E(Q₃) + ... .

Field lines

Field lines were introduced by Michael Faraday to help visualize the direction and magnitude of the electric field.  The direction of the field at any point is given by the direction of a line tangent to the field line, while the magnitude of the field is given qualitatively by the density of field lines.  The field lines converge at the position of a point charge.  Near a point charge their density becomes very large.  The magnitude of the field and the density of the field lines scale as the inverse of the distance squared.  

Field lines start on positive charges and end on negative charges.

Rules for drawing field lines:

• Electric field lines begin on positive charges and end on negative charges, or at infinity.
• Lines are drawn symmetrically leaving or entering a charge.
• The number of lines entering or leaving a charge is proportional to the magnitude of the charge.
• The density of lines at any point (the number of lines per unit length perpendicular to the lines themselves) is proportional to the field magnitude at that point.
• At large distances from a system of charges, the field lines are equally spaced and radial as if they came from a single point charge equal in magnitude to the net charge on the system (presuming there is a net charge).
• No two field lines can cross, since the field magnitude and direction must be unique.

Examples:

The field lines of an electric dipole, i.e. a positive and a negative charge of equal magnitude, separated by a distance d.
The electric field decreases with distance as 1/(distance)³, much faster than the field of a point charge.

The field lines of two positive charges of equal magnitude separated by a distance d.
The electric field decreases with distance as 1/(distance)².

Polar molecules do not have a net charge, but the centers of the positive and negative charge do not coincide.  Such molecules produce a dipole field and interact via the electrostatic force with their neighbors.

Example:

Every water molecule (H₂O) consists of one oxygen atom and two hydrogen atoms.  A water molecule has no net charge because the number of positively charged protons equals the number of negatively charged electrons in each atom.  In the water molecule, each hydrogen atom is bound to the oxygen atom by a covalent bond.  The hydrogen atom and the oxygen atom share an electron, which has a slightly higher probability to be closer to the oxygen atom than to the hydrogen atom.  This results in a polar molecule, in which the oxygen "side" of the molecule has a slight negative charge, while the hydrogen "sides" have a slight positive charge.  Because water is a polar molecule it dissolves many inorganic and organic materials.  Organisms can build macromolecules to attract or repel water as needed simply by varying the charge on side chains.

Problem:

The figure on the right shows the electric field lines for a system of two point charges.
(a) What are the relative magnitudes of the charges?
(b) What are the signs of the charges?

Solution:

• Reasoning:
  Rules for drawing field lines:
  Electric field lines begin on positive charges and end on negative charges, or at infinity.
  The number of lines entering or leaving a charge is proportional to the magnitude of the charge.
• Details:
  (a) There are 32 lines coming from the charge on the left, while there are 8 converging on that on the right.  Thus, the magnitude of the charge on the left is 4 times larger than the magnitude of the charge on the right.
  (b) The charge on the left is positive, the charge on the right is negative.

Please watch this video clip that helps you "see" the electric field!

Embedded Question 2

Which of the diagrams below is not a valid field line diagram, and why?

Discuss this with your fellow students in the discussion forum!
Review the rules for drawing electric field lines.

The electric field exists in 3-dimensional space.  Below you can explore 3D drawings of the electric field lines of two charges.  Please click on the image!
You can change the magnitude (arb. units), sign, and position (arb. units) of the each charge.  You can use the mouse to change your viewing angle and zoom in or out.  To view field lines of one charge only, choose zero for the magnitude of charge B.

Usually we try to visualize the magnitude and direction of the electric field by drawing field lines in a plane in two dimensions.  For two or three charged spheres lying in a plane, we could draw the projections of the field lines onto this plane.  But that would make it impossible to represent the exact direction of each line.  We could also only draw field lines that lie in that plane.  Then the directions of the drawn lines would be correct, but we would missing most of the lines, and the number of lines would not be proportional to the magnitude of the charge.  So we make a hybrid 2D drawing.   For the directions, we only choose field lines lying in the plane, but we scale the number of these lines up to the number we would get from a projection.  Therefore, for charged spheres lying in a plane, the field strength in a 2D drawing is not proportional to the density of the field lines, but to the square of the density of the lines.

Forces and fields due to continuous charge distributions

Even though charge is quantized, we often can treat it as being continuously distributed inside some volume, since one quantum of charge is a tiny amount of charge.  On a macroscopic scale we define the volume charge density ρ = lim_ΔV-->0(ΔQ/ΔV) as the charge per unit volume, the surface charge density σ = lim_ΔA-->0(ΔQ/ΔA) as the charge per unit area, and the line charge density λ = lim_ΔL-->0(ΔQ/ΔL) as the charge per unit length.

We then have for the electric field of a distribution of charges in a volume V

E(r) =  [1/(4πε₀)]∑_iq_i(r - r_i)/|r - r_i|³] --> [1/(4πε₀)][∫_v' dV' ρ(r')(r - r')/|r - r'|³.

Here (r - r_i)/|r - r_i| is the unit vector pointing from r_i to r, and (r - r')/|r - r'| is a unit vector pointing from the volume element dV' at r' to r.

If the charge is distributed over a surface, then ρdV --> σdA, where σ is the surface charge density and dA is an element of surface area.  For a line charge distribution we have ρdV --> λdl, where λ is the line charge density and dl is an element length.

Problem:

Consider a line charge with line charge density λ = Q/2a that extends along the x-axis from x = -a to x = +a.  Find the electric field on the y-axis.

Solution:

• Reasoning
  We are asked to find the electric field due to a line charge distribution.  This is a continuous charge distribution.
  We find the he field on the y-axis due to an infinitesimal element of charge λdx at position x and the integrate over all positions x from x = -a to x = +a.
• Details of the calculation:
  
  
  The field on the y-axis due to an infinitesimal element of charge λdx is given by
  dE_x = (k_eλdx/(x² + y²)) cosθ,   dE_y = (k_eλdx/(x² + y²)) sinθ,
  where cosθ = x/(x² + y²)^½, sinθ = y/(x² + y²)^½.
  
  On the y-axis the field due to the line charge therefore is given by
  E_x = k_eλ∫_-a^axdx/(x² + y²)^3/2 = 0 from symmetry, and
  E_y = k_eλy∫_-a^adx/(x² + y²)^3/2.
  Using ∫_-a^adx/(x² + y²)^3/2 = 2a/(y²(a² + y²)^½)
  we have  E_y = k_eλ 2a/(y(a² + y²)^½) = k_eQ/(y(a² + y²)^½)
  
  The electric field on the y-axis points in the positive y-direction for y > 0 and in the negative y-direction for y < 0.
  As y becomes very large, we can neglect a² compared to y² under the square root and then E_y = k_eQ/y² ∝ 1/y².  From very far away, the line charge looks like a point charge.
  If, on the other hand, the line is very long, and we y << a, then we can neglect y² compared to a² under the square root and then E_y = k_eQ/(ay) ∝ 1/y.  Near a very long line charge the electric field falls off as 1/distance, not as 1/distance².

Simulation:  Electric Field Hockey 

• This simulation looks down on an air hockey table.  Instead of hitting the puck, you move it using charges.
• Use the Practice mode for testing your ideas.
• Clear zeros everything, and Reset brings the puck back to the starting point with the same charges.
• You can change the sign of the charge on the puck by checking or un-checking the Puck is Positive option.
• You may want to investigate how to use multiple charges to make a goal.

External link to other web material:  The Physics Classroom: Static Electricity