U1: Electrostatics

What Is Electrostatics?

Electrostatics is the study of charges at rest. It is weighted at about 26-34% of the exam. We study charges at rest (that aren't moving relative to each other) in this unit only because it simplifies the learning a bit. When charges move, they generate a magnetic field as well. This complicates stuff beyond what we're concerned with for the moment, so for this unit you only consider charges that are at rest or moving very slowly.

Charged Particles Review

Here’s a quick review on charged particles:

● Protons carry a positive charge.

● Electrons carry a negative charge.

● An object with a net charge has an excess of protons or electrons.

● Similar charges repel, opposite charges attract.

Charge is measured in the unit of coulombs. A single proton has a charge of +e, while an electron has a charge of -e. e is the elementary charge, defined as the electric charge a proton carries, and is about 1.602 × 10-19 coulombs. Charged particles exert force on other charged particles, with the direction determined by their charge (same or opposite to each other).

The Electrostatic Force

The electrostatic force that a point charge of q1 would exert on another charge q2 is given by

Coulomb’s Law for Electrostatic Force:

𝐹12 = (𝑘𝑞1𝑞2)/(|𝑟12|)2 * (𝑟̂12)

Here’s what the symbols mean:

● F12 is the force that charge q1 exerts on charge q2, in Newtons.

● k is Coulomb’s constant, about 8.99 × 109 N m2/C2.

● q1 and q2 are charges 1 and 2 being considered, in coulombs.

● r12 is the distance vector from q1 to q2.

o |r12|2

is the square of the scalar distance between charges 1 and 2.

o 𝑟̂12 is the unit vector pointing from charge 1 to 2. This partially determines the

direction of the electrostatic force that charge 1 will exert.

If the vector notation scares you, here’s the scalar form that often also works:

𝐹12 = 𝑘𝑞1𝑞2/𝑟2

Electrostatic Force With Multiple Charges

When you have multiple point charges that are close to each other, each of them is exerting an electrostatic force on each other. To find the total electrostatic force F exerted on charge Q by N charges qi, we simply do a vector sum of all the forces.

𝐹 = ∑𝑖 𝐹i = 𝑘𝑄 (𝑁∑𝑖=1 (𝑞1)/(𝑟2i) * 𝑟̂i)

If the summation notation looks scary, don’t fret. This is just a fancy way of writing “sum up each of the force vectors from each charge to find the net force.”

Electrostatic Force With Objects That Take Up Space

The electrostatic force equations that we just went over only cover interactions between point charges. A point charge is a charge that doesn’t take up any space: it’s a single point with some charge. When you have a charged object with spatial extent (that takes up space), however, you have to approach it a little differently. Let’s go back to our sum of forces from before.

𝐹 = ∑𝑖 𝐹𝑖 = 𝑘𝑄 (𝑁∑𝑖=1 (𝑞1)/(𝑟2i)𝑟̂i)

We can think of charged objects that take up space as a massive collection of an infinite number of small charges. Now, what would happen if we had an infinite amount of tiny charges, e.g. 𝑁 →∞? Well, we get:

𝐹 = 𝑘𝑄 ∞∑𝑖=1 (𝑞1)/(𝑟2i)𝑟̂i

If you’ve done some calculus, you’ll know that this looks suspiciously like it can be represented with an integral. If you thought that, you’d be right!

𝐹 = ∫ 𝑑𝐹 = 𝑘𝑄 ∫ 𝑑𝑞/(𝑟2) * 𝑟̂

So, to solve a problem like the one below, all you need to do is set up the right equations and integral and solve. Try finding the force exerted by the charged rod Q on q.

Electric Field

What Is An Electric Field?

The electric field E created by a charge q_1 is a vector function called a vector field, that shows how the charge affects other charges around it. It is very similar to the concept of the gravitational field generated by objects with mass.

Electric Field Near A Point Charge

The electric field a distance r away from point charge q is given by:

𝐸(𝑞, 𝑟) = 𝑘𝑞/(|𝑟|2) * 𝑟̂

The direction of E is radially outward from a positive point charge and radially inward towards a negative charge.

Once again, if you don’t like or need the vector notation, you can use the scalar form of the equation.

𝐸(𝑞, 𝑟) = 𝑘𝑞/(𝑟2)

To find the electrostatic force exerted on a charge due to the electric field it’s in, multiple E and the charge q, just like you would with gravity.

𝐹q = 𝐸𝑞 ⟶ 𝐹g = 𝑚𝑔

See the similarity?

Electric Field Lines

Electric field lines visually describe an electric field. The lines in an electric field line diagram describe the direction a positive test charge would accelerate if placed where the line was.

They begin from a charge and end at infinity, and never intersect. They also don’t have any ends: they extend out to infinity.

They point outwards from positive charges and inwards from negatives charges.

Electric Field With More Than One Charge

The notation and thinking for electric fields with multiple charges is the same as with electrostatic force: simply sum up the fields created by each charge. This is the equation for the electric field at any position r.

𝐸 = ∑𝑖 𝐸i = 𝑘{∑𝑖 (𝑞i)/(𝑟2i) * 𝑟̂i}

Electric Field Of Charges With Spatial Extend

Like with electrostatic force, we can conceptualize charged objects that take up space as an infinite amount of small point charges. Therefore, we can apply the same method to turn the infinite sum into an integral. We get the following equation.

𝐸 = ∫ 𝑑𝐸 = 𝑘 ∫ (𝑑𝑞)/(𝑟2) * 𝑟̂

This integral isn’t always trivial. However, they're often ways you can simplify your equations and your integral to make your life easier, like looking for symmetry. Check out this HyperPhysics page to see an example of this integral in use.

Something To Remember

Electric fields, like gravitational fields, don’t do anything until a charge enters the field. The field is a description of how a charge will influence other charges, so if there aren’t any other charges, then the electric field isn’t actually doing anything at all.

Gauss’s Law And Electric Flux

What Is Flux?

Flux is a concept that is important to many areas of physics. The flux of a vector quantity X is the amount of the quantity flowing through a surface.

The direction of infinitesimal area dA is outward normal to the surface.

Electric Flux

Flux can be of something physical, like water, or of something abstract, like an electric field, which is what we are looking at right now. You can compute a flux with a surface and a vector field X = X(x,y,z). With electric flux, our vector field X is just referring to the electric field E.

𝛷_𝑞 = ∫ 𝐸 ⋅ 𝑑𝐴

Here, we are taking the dot product of the field with the “area vector”, to get the amount of the vector pointing in the direction perpendicular to the surface. (This is equivalent to the 𝐸𝐴𝑐𝑜𝑠(𝜃) in the diagram above: that one just uses the angle instead of the dot product.)

Gauss’s Law

Gauss’s Law tells us the electric flux if we have a closed surface, like a sphere or cube. The formula

is as follows:

𝛷𝑞 = ∮ 𝐸 * 𝑑𝐴 = 𝑄encl/𝜖0

The scary integral symbol with a circle in it is a surface integral: it means you are adding up the infinitesimal bits over the surface you are considering. Q_encl is the charge enclosed by the closed surface, while 𝜖0 is the permittivity of free space, about 8.85 * 10-12𝐶2/𝑁𝑚2. This is basically how easily an electric field can permeate in a vacuum.

You remember that constant mentioned in the previous 2 chapters, k? Well, it’s quite closely related to the permittivity of free space:

𝑘 = 1/4𝜋𝜖0

Now let’s figure out why. Imagine a charge q that is surrounded by a sphere of radius r. Let’s try calculating the electric flux.

𝛷𝑞 = ∮ 𝐸 ⋅ 𝑑𝐴 = 𝑄encl/𝜖0

𝐸𝐴 = 𝑞/𝜖0 ⟶ 𝐸 = 𝑞/𝐴𝜖0

𝐴 = 4𝜋𝑟2 ⟶ 𝐸 = (1/4𝜋𝜖0)(𝑞/(𝑟2)) = 𝑘𝑞/𝑟2

Pretty interesting, huh? Flux is already showing us new things. Let’s go on.

Electric Potential Energy

The work done by any force is 𝑊 = ∫ 𝐹 ⋅ 𝑑𝑟. Let’s try taking the integral of the electrostatic force!

𝑊 = ∫𝑟2𝑟1  𝐹𝑞 * 𝑑𝑟 = 𝑘𝑞1𝑞2  ∫𝑟2𝑟1  𝑑𝑟/(𝑟2) = −𝑘𝑞1𝑞2/(𝑟)|𝑟2𝑟1 = −𝛥𝑈𝑞

So, electric potential energy is given by

𝑈q = 𝑘𝑞1𝑞2/𝑟

Now, let’s consider what happens when we have positive/negative charges.

• If both charges are positive or negative, Uq will be positive. This means that it takes positive work to bring to 2 charges together from 𝑟 = ∞ to 𝑟. Makes sense, right? When the 2 charges are both positive or negative, they will repel, which means you will have to do work to bring them together.

• If 1 charge is positive and the other negative, Uq will be negative. This means that it will take negative work to bring the 2 charges together. This also makes sense, intuitively: the 2 charges will be attracting each other!

Electric Potential (Not Energy)

Electric potential is electrical potential energy per unit charge and is measured in units of joules per coulomb. For a charge q in a field created by q_source:

𝑉 = 𝑈q/𝑞 = 𝑘𝑞s/𝑟

Electric potential is measured in volts (joules per coulomb: 1V = 1 J/C). When a charge of q and a charge of 2q are displaced in the same way from one point to another in an electric field, in both cases, the ratio of the change in potential energy to the charge being displaced is equal is to the change in electric potential (the electric potential difference) from the first point to the endpoint.

Electric Potential Difference

The change in electric potential is called the electric potential difference, or voltage. It’s also measured in volts (joules per coulomb).

𝑉 = 𝑈q/𝑞

↓

𝛥𝑉 = 𝛥𝑈q/𝑞 and 𝑑𝑉 = 𝑑𝑈q/𝑞

You can also represent electric potential difference as an integral.

𝛥𝑈q = − ∫ 𝐹 * 𝑑𝑟 ⟶ 𝛥𝑈q/𝑞 = 𝛥𝑉 = − ∫  𝐹/𝑞 * 𝑑𝑟

𝛥𝑉 = − ∫  𝐸 * 𝑑𝑟

Equipotential Lines

Equipotential lines are lines perpendicular to the electric field lines, where the electric potential is the same anywhere on the line.

Because the electric potential anywhere on an equipotential line is the same, no work is done when moving between points on an equipotential line.

Weird questions sometimes come up involving this: remember it.

Electric Field Near An Infinite Plane Of Charge

Let’s talk about the field near an infinitely large plane of charge. It’s got a charge density of 𝜎. By symmetry, the field E must be perpendicular to the plane. We have a Gaussian surface (in this case, a cylinder) with base area A. The height doesn’t really matter.

Let’s try and calculate flux: nothing is flowing out of the side, only out the ends of the cylinder.

The flux is therefore 𝛷𝑞 = 𝐸(2𝐴).

We can then use Gauss’s Law.

∮ 𝐸 * 𝑑𝐴 = 𝑄encl/𝜖0 = 𝜎𝐴/𝜖0

𝜎𝐴/𝜖0 = 𝐸(2𝐴) ⟶ 𝐸 = 𝜎/2𝜖0

We can see that E is a constant, independent of distance from the plane. Also, both sides of the plane are the same.

Electric Field Between 2 Parallel Charged Plates

Let’s imagine 2 parallel plates of uniform charge density, with electric field flowing in the same direction between them, separated by distance d with a potential difference of 𝛥𝑉. What is the electric field inside the 3 plates?

Because it’s like 2 charged planes, the electric field between the 2 plates will be uniform.

Imagine the work needed to move a charge q from the positive plate (call it plate A) to the bottom plate (call it plate B).

𝑊 = 𝛥𝑈q = 𝑞𝛥𝑉

The potential difference VAB between A and B is

−𝛥𝑉 = −(𝑉B − 𝑉A) = 𝑉A − 𝑉B = 𝑉AB.

Work is also 𝑊 = 𝐹𝑑 because of the constant field. 𝐹 = 𝐸𝑞, so 𝑊 = 𝐸𝑞𝑑.

𝐸𝑞𝑑 = 𝑞𝑉AB ⟶ 𝐸 = 𝑉AB/𝑑

Again, we have a field that is the same, this time throughout the area between the plates.