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Kurs: Elektrotechnika > Rozdział 5

Lekcja 1: Electric force and electric field

Electric force

Electric force exists between charges, as described by Coulomb's Law. Worked example: a line of charge with q off the end. Written by Willy McAllister.

Our study of electricity begins with electrostatics and the electrostatic force, one of the four fundamental forces of nature. Electrostatic force is described by Coulomb's Law. We use Coulomb's Law to solve the forces created by configurations of charge.

Electrostatics deals with forces between charges. Static means the charges are not moving, or at least are not moving very fast.

How fast is "not very fast"?

When a charge moves, it generates a magnetic field, which can lead to a magnetic force. As we begin our study of electricity, we want to minimize complexity and focus just on electric force. To keep things simple, we let charges stand still, or move very slowly. So "not very fast" means motion so slow the magnetic effects are insignificant compared to the electric forces.

Ładunek elektryczny

How do we know there is such a thing as charge? The concept of charge arises from an observation of nature: We observe forces between objects. Electric charge is the property of objects that gives rise to this observed force. Like gravity, electric force "acts at a distance". The idea that a force can "act at a distance" is pretty mind-blowing, but it's what nature really does.

Electric forces are very large, far greater than the force of gravity. Unlike gravity, there are two types of electric charge, (whereas there is only one type of gravity; gravity only attracts).

Unlike charges attract,

Like charges repel,

Force between charges: Coulomb's Law of electric force

Coulomb's Law very nicely describes this natural phenomenon. The law has this form,

\vec{F} = K \frac{q_{0} q_{1}}{r^{2}} \hat{r}

Where

$\vec{F}$ ‍ is the electric force, directed on a line between the two charged bodies.
$K$ ‍ is a constant of proportionality that relates the left side of the equation (newtons) to the right side (coulombs and meters). It is needed to make the answer come out right when we do a real experiment.
$q_{0}$ ‍ and $q_{1}$ ‍ represent the amount of charge on each body, in units of coulombs (the SI unit for charge).
$r$ ‍ is the distance between the charged bodies.
$\hat{r}$ ‍ is a variable unit vector that reminds us the force points along the line between the two charges. If the charges are alike, the force is repulsive; if the charges are unlike, the force is attractive.

The Electric Constant, $ϵ_{0}$ ‍, the permittivity of free space

K

, the constant of proportionality, frequently appears in this form,

This notation is motivated by a theory we come across later, Gauss's Flux Theorem. There, we discover the constant

K

contains a geometric factor and another factor contributed by the experimental setup. Gauss's Theorem involves the surface area of a sphere. The geometric factor

4 π

is the solid angle of a sphere (analogous to the

2 π

total angle for a circle). The rest of the proportionality constant represents what the experimental setup gives us, captured in the

ϵ_{0}

term. This notation results in a simple expression of Gauss's Law. Teachers think it is best to bring this form of the constant all the way back here to the beginning of the study of electricity where we often see it in Coulomb's Law.

K = \frac{1}{4 π ϵ_{0}}

and Coulomb's Law is written in this form,

\vec{F} = \frac{1}{4 π ϵ_{0}} \frac{q_{0} q_{1}}{r^{2}} \hat{r}

The Greek letter

ϵ_{0}

is the electric constant, also known as the permittivity of free space, (free space is a vacuum). Coulomb's Law describes something that happens in nature. The electric constant,

ϵ_{0}

, describes the experimental setup and the system of units. "Experimental conditions" refers measuring

\vec{F}

on point charges (or something that acts like a point charge, like charged spheres). In the SI system of units,

ϵ_{0}

is experimentally measured to be,

ϵ_{0} = 8,854 187817 \cdot 10^{- 12}

coulomb

^{2} /

newton-meter

^{2}

This value of

ϵ_{0}

makes,

K = \frac{1}{4 π ϵ_{0}} = \frac{1}{4 π \cdot 8,854 \cdot 10^{- 12}} = 8,987 \cdot 10^{9}

or for engineering purposes, we round

K

to something easier to remember,

K = \frac{1}{4 π ϵ_{0}} = 9 \cdot 10^{9}

The dimensions of

K

are: newton-meter

^{2} /

coulomb

^{2}

Example: three point charges

For our first example we use Coulomb's Law to compute the force on a charge from two nearby charges. We set up three charges on the vertices of a

30^{\circ} - 60^{\circ} - 90^{\circ}

triangle.

q_{2}

, with the dark outline, is our test charge.

Now assign some values to the charges (coulombs) and spacing (meters),

Find the force (magnitude and direction) on $q_{2}$ ‍, the $+ 3 C$ ‍ charge.

Compute the force between each pair of charges. In this example there are two force vectors to think about, {

q_{0}

q_{2}

}, and {

q_{1}

q_{2}

}. The individual force vectors are on a direct line between the charge pairs.

For simplicity, we'll use

K

as the proportionality constant. Apply Coulomb's Law to compute the force. We manage the magnitudes and angles separately. The magnitudes of the forces are,

F = K \frac{q_{0} q_{1}}{r^{2}}

F_{02} = K \frac{4 \cdot 3}{(\sqrt{3})^{2}} = K \cdot 4

force on

q_{2}

from

q_{0}

(repels)

F_{12} = K \frac{1 \cdot 3}{(1)^{2}} = K \cdot 3

force on

q_{2}

from

q_{1}

(attracts)

We have solved the magnitudes of the pairwise forces.
The final step is to perform a vector sum to get the magnitude and direction of the final force vector.

The force vectors form the sides of a 3-4-5 right triangle.
The magnitude of the resultant force is,

| F_{2} | = K \cdot \sqrt{3^{2} + 4^{2}} = K \cdot 5

Figure out angle

∠ {\vec{F}}_{2}

by counting degrees from horizontal, starting at the

4 C

charge,

Interior angles of our two triangles,

The angles of the 3-4-5 triangle come from,

\arcsin (4 / 5) = 53, 13^{\circ}

and

\arcsin (3 / 5) = 36, 86^{\circ}

Merging the triangles together shows how the angles combine (blue arrows):

The

30^{\circ}

angle gets a negative sign because it is rotating clockwise, while the

36, 9^{\circ}

angle adds with a positive sign because it is rotating counterclockwise.

∠ {\vec{F}}_{2} = - 30^{\circ} + 36, 9^{\circ} = + 6, 9^{\circ}

Combining the magnitude and angle, the force

{\vec{F}}_{2}

q_{2}

in newtons is,

\vec{F_{2}} = K \cdot 5 ∠ 6, 9^{\circ}

\vec{F_{2}} = (9 \cdot 10^{9}) \cdot 5 ∠ 6, 9^{\circ}

\vec{F_{2}} = 4, 5 \cdot 10^{10} ∠ 6, 9^{\circ} newtons

Example: line of charge with a point charge off the end

A line of charge

L

meters long has a total charge of

Q

. Assume the total charge,

Q

, is uniformly spread out on the line. A point charge

q

is positioned

a

meters away from one end of the line.

Find the total force on a charge $q$ ‍ positioned off the end of a line of charge.

The line contains a total charge

Q

coulombs. We can approach this problem by thinking of the line as a bunch of individual point charges sitting shoulder to shoulder. To compute the total force on

q

from the line, we sum up (integrate) the individual forces from each point charge in the line.

We define the charge density in the line as

\frac{Q}{L}

coulombs/meter.

The idea of charge density lets us express the amount of charge,

d Q

, in a little piece of the line,

d x

, as,

d Q = \frac{Q}{L} d x

d Q

is close enough to being a point charge to allow us to apply Coulomb's Law. We can figure out the direction of the force right away: The force on

q

from every

d Q

is directed straight between

q

and

d Q

. Direction solved, now the magnitude of the force,

d F = \frac{1}{4 π ϵ_{0}} \frac{q d Q}{x^{2}}

The numerator multiples the two charges,

q

and

d Q

; the denominator

x

is the distance between the two charges.

To find the total force, add up all the forces from each little

d Q

's by integrating from the near end of the line (

a

), to the far end (

a + L

F = \int_{a}^{a + L} d \vec{F} = \int_{a}^{a + L} \frac{1}{4 π ϵ_{0}} \frac{q d Q}{x^{2}}

This equation includes both

x

and

d Q

as variables. To get down to a single independent variable, eliminate

d Q

by replacing it with the expression

Q / L d x

from above,

F = \int_{a}^{a + L} \frac{1}{4 π ϵ_{0}} \frac{q Q}{L} \frac{1}{x^{2}} d x

Move everything that does not depend on

x

outside the integral.

F = \frac{1}{4 π ϵ_{0}} \frac{q Q}{L} \int_{a}^{a + L} \frac{1}{x^{2}} d x

And solve the integral,

The indefinite integral of a power of

x

\int x^{n} d x = \frac{x^{n + 1}}{n + 1}

, for any

n \neq - 1

. In this example,

n = - 2

\int_{a}^{a + L} x^{- 2} d x = \frac{x^{- 2 + 1}}{- 2 + 1} = \frac{x^{- 1}}{- 1} = - \frac{1}{x}

Apply the limits and solve the definite integral,

- \frac{1}{x} |_{a}^{a + L} = - [\frac{1}{(a + L)} - \frac{1}{a}] = \frac{1}{a} - \frac{1}{(a + L)}

Multiply each term by an identity and combine fractions,

\frac{1}{a} (\frac{a + L}{a + L}) - \frac{1}{(a + L)} (\frac{a}{a}) = \frac{(a + L) - a}{a (a + L)}

subtract the

a

's in the numerator to end up with solution to the definite integral,

\int_{a}^{a + L} x^{- 2} d x = \frac{L}{a (a + L)}

Substitute this solution to the definite integral back into the force equation,

\vec{F} = \frac{1}{4 π ϵ_{0}} \frac{q Q}{L} \frac{L}{a (a + L)}

leaving us with the final force equation for a line of charge and a point charge off the end ...

F = \frac{1}{4 π ϵ_{0}} \frac{q Q}{a (a + L)}

Some things to notice about the solution:

The numerator is the product of the test charge and the total charge on the line, which makes sense.
The denominator has the form ${distance}^{2}$ ‍, created by a combination of distance to the near end and far end of the line. The $a (a + L)$ ‍ form of the denominator emerges from the particular geometry of this example.
If the point charge $q$ ‍ moves very far away from the line, $L$ ‍ becomes insignificant compared to $a$ ‍, and the denominator approaches $a^{2}$ ‍. So at great distance, the line starts to resemble a far-off point charge, and as one would hope, the equation approaches Coulomb's Law for two point charges.

We'll do a few more electrostatics problems with simple charge geometries. After that, the math gets really involved, so the common strategy with complex geometries becomes: break down the geometry into simpler versions we already know how to do, then merge the answers.

Strategies for applying Coulomb's Law

Coulomb's Law is a good choice for situations with point charges and/or simple symmetric geometries like lines or spheres of charge.

Since Coulomb's Law is based on pairwise forces between charges, when faced with multiple (more than two) point charges,

Work out the forces between each pair of charges.
Finish with a vector addition to merge the pairwise forces into a single resultant force.

For a situation with distributed charge, creatively model the distributed charge as a collection of point charges,

Invent a little $d Q$ ‍ representing an infinitesimal charge within the region of distributed charge.
Work out the forces pairwise between the point charge and each little $d Q$ ‍.
Sum up the forces with an integral. This is a vector sum to get the resultant force.

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Kurs: Elektrotechnika > Rozdział 5

Electric force

Ładunek elektryczny

Force between charges: Coulomb's Law of electric force

The Electric Constant, $ϵ_{0}$ ‍, the permittivity of free space

Example: three point charges

Find the force (magnitude and direction) on $q_{2}$ ‍, the $+ 3 C$ ‍ charge.

Example: line of charge with a point charge off the end

Find the total force on a charge $q$ ‍ positioned off the end of a line of charge.

Strategies for applying Coulomb's Law

Related

References

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Elektrotechnika

Kurs: Elektrotechnika > Rozdział 5

Ładunek elektryczny

Force between charges: Coulomb's Law of electric force

The Electric Constant, ϵ0‍ , the permittivity of free space

Example: three point charges

Find the force (magnitude and direction) on q2‍ , the +3C‍ charge.

Example: line of charge with a point charge off the end

Find the total force on a charge q‍ positioned off the end of a line of charge.

Strategies for applying Coulomb's Law

Chcesz dołączyć do dyskusji?

The Electric Constant, $ϵ_{0}$ ‍, the permittivity of free space

Find the force (magnitude and direction) on $q_{2}$ ‍, the $+ 3 C$ ‍ charge.

Find the total force on a charge $q$ ‍ positioned off the end of a line of charge.