Comprehensive Guide to Coulomb's Law with Practical Problem Solutions

in this video we're going to focus on coulomb's law we're also going to work on some

problems associated with it but before we do that let's begin our discussion with the atom

now at the center of an atom you have the nucleus and inside the nucleus of an atom there

are subatomic particles known as protons and neutrons neutrons are neutral in charge

protons carry a positive charge whereas electrons which orbit the nucleus

they carry negative charges one proton has a charge of 1.6 times 10 to the negative 19 coulombs

since we don't have fractional protons electric charge is said to be quantized it exists in discrete amounts

so this is the fundamental value of charge so electric charge will always be a

multiple of this number now an electron has the same magnitude of charge but with a negative sign

so the charge of an electron is negative 1.6 times 10 to the negative 19 clues

now let's place a positive charge next to a negative charge what's going to happen if we

put these two charges next to each other let's call the first charge q1 and the second charge q2

we need to know is that opposite charges attract like charges repel

so because these two charges are opposite in sign they're going to attract each other

q1 is going to fill a force that will basically

propel it towards q2 and q2 will also feel the same force these two forces are equal in magnitude

but opposite in direction now the equation that helps us to calculate that force also known as an

electric force is klum's law the electric force is equal to

the proportionality constant k times the magnitude of q1 times q2 divided by r squared

now r is the distance between the center of these two charges

now these charges are known as point charges where the size of the charge is

irrelevant the proportionality constant k is equal to

9 times 10 to the 9

newtons over square meters times square coulombs so make sure you know this value because

we're going to use it when we're solving problems now this proportionality constant is

always equal to i mean it's also equal to one over four pi

times epsilon naught and epsilon in this

problem or in this situation is equal to 8.85 times 10 to the negative 12

gloom squared over newton's times square meters so this is the permittivity of free

space now when solving problems associated with columbus law

k is the value that you're going to use most often now we said that

opposite charges attract let's draw a picture with two like charges

so if we were to put a positive charge next to another positive charge these two charges will repel

so they will fill a force that will push them apart but these two forces are equal in

magnitude but opposite in direction so now to review

remember protons carry a positive charge and electrons they carry

a negative charge now whenever you have an object that has a net positive charge

you need to be aware that what it means that is that there's more protons than electrons in that object

if you have an object that is negatively charged that means that it has more electrons

than protons so whenever the number of protons and electrons are equal

the object will be neutral in charge but if you have more protons than electrons

the object will carry a positive charge if there are more electrons and protons then it's going to have a negative

charge now let's go back to coulomb's law this equation describes the relationship

between the electric force the magnitude of the charges and the

distance between the two charges if we were to increase the magnitude of one of the charges

the electric force will increase so let's say this is q1

and this is let's put a positive charge and let's say q2 also has a positive charge

and let's say the force acting on each of them is 100 newtons now what do you think is going to happen

if we increase the magnitude of one of those charges let's say

we double the value of q2 so it's two times q2 well since we increase the

magnitude of the charge the electric force is going to increase in value it's going to double it's going

to become 200 newtons so if you double the magnitude of one of the charges the force will double

now let's say this is r what happens if we

place q2 further away from q1 so let's say if we double the distance

between q q1 and q2 so this is going to be 2r instead of 1r what's going to happen to the force

notice that r is on the bottom of the fraction if you were to increase the distance

between the two charges the magnitude of the force will decrease the force

is inversely related to the square of the distance so the force is not not going to cut in

half it's going to be one-fourth of its original value because if you were to plug in two

into that equation and one for everything else it would be one over two squared which is one over four

so it's going to be one-fourth of its original value which means it's going to be 25 newtons

so if you were to double the magnitude of one of the charges the force will double

if you were to double the distance between the two charges the force will be reduced by a factor of

four so the electric force is strongest when

the charges are placed closer together or if you increase the magnitude of the charges

now let's talk about the units associated with coulomb's law the unit of force is the newton

now for q1 or q2 which represents the electric charge the unit for that is coulombs

one coulomb is a large amount of charge so typically you'll see the charge represented in microclones

one micro coulomb is one times ten to the minus six coulombs one milli cloon

is one times ten to the negative three coulombs and one nano clue

is one times ten to negative nine clumps now r the distance between the charges needs to be in meters

sometimes you'll see values in centimeters or millimeter so remember

one centimeter is one times ten to the minus two meters and one millimeter

is one times ten to the minus three meters you could also see it this way

one meter is equivalent to a hundred centimeters and one milli i mean one meter is

equivalent to a thousand millimeters so those are some other things that you

want to know uh when dealing with problems associated with klum's law

now let's work on some practice problems let's start with this one a 10 micro coulomb point charge

is 25 centimeters away from a negative 20 micro coulomb point charge calculate the magnitude of the electric

force between them so let's begin by drawing a picture so we have two charges

one it's positive the other is negative we'll call this q1 and q2

so q1 has a magnitude of positive 10 micro coulombs q2 has a magnitude of negative 20

micro coulombs and the distance between the two charges is 25 centimeters

so now let's calculate the force acting on the two charges now keep in mind since we're dealing

with opposite charges these two will feel a force of attraction

so these two forces will be equal in magnitude the force

that is acting on charge one due to charge two you can call it f one comma two

the force that is acted on charge 2 due to charge 1 you can call it f 2 comma 1 but for this problem they're

exactly the same now the proportionality constant k is nine times ten to the nine

q one is ten micro coulombs so to plug this in replace micro coulombs with

ten to the negative six coulombs q two is negative twenty 20 micrograms now for this formula you don't need to

worry about the negative sign because we already know the direction of the force

also we only want the magnitude of the electric force

electric force is a vector it has both magnitude and direction but since we only want the magnitude

we don't need to be concerned with the direction and we don't need to worry about the negative sign

so we're going to plug in positive 20 times 10 to the minus 6 coulombs for q2

now r the distance between the two charges is in centimeters we need to convert that

to meters well actually you can simply plug it in like this

25 centimeters you can write it as 25 times 10 to the minus 2 meters you can replace centi with 10 to

negative 2. but if you wanted to convert it here's what you could do

you can multiply 25 centimeters by one meter over a hundred centimeters and this will give you

0.25 meters when you divide 25 by 100 or you can move the decimal point two

units to the left that will give you 0.25 meters which is the same as 25 times 10 to the negative 2.

so let's go ahead and plug everything in nine times ten to the nine times ten

times ten to the minus six times the other number divided by

and for this part you want to put it in parentheses you can divide it by 0.25 squared if you want to

or type in exactly what you see here so the answer that you should get

is 28.8 newtons so that is the magnitude of the electric force

that is acting on both of the two charges now let's move on to number two

the electric force between two point charges with a magnitude of

800 nano columns and 900 nano columns is 15 units how far apart are the two charges from

each other in centimeters so we can draw another picture if we want to so here's the first charge and

here's the second charge both of the two charges are positive we need to calculate the distance

between them and we know the force acting on each of them

now because these two have the same charges they will repel each other with a force

of 15 newtons but how can we calculate r well let's start with columbus law

f is equal to k q 1 over q 2 i mean k q 1 times q 2 over r squared so what we're going to do is

we're going to isolate r in this equation so to do that let's multiply both sides

by r squared first so these will cancel and we're going to get

f times r squared is equal to k q1 q2 now let's divide both sides by f

so this will give us r squared is equal to k q1 q2 over f

now to get r by itself we need to take the square root of both sides so this is the formula that we could use

to calculate the radius or not really the radius but i think of the radius when i see r but

the distance between the two charges so r is going to equal the square root of k

q 1 times q 2 divided by f so k is nine times ten to the nine

q one it really doesn't matter which one is q one so let's just make this one q one

and the other one q two q minus eight hundred nano coulombs given my nano

is 10 to the minus 9 q2 is 900 nano columns so 900 times 10 to the

minus 9 and then f is 15 newtons now for those of you who want to see the

units k i'm going to plug in the units in this

formula k has the units newtons over

square meters square coulombs q1

is in clones q2s and clones f is in newtons so the unit newtons cancel

the unit coulombs cancels so we have square meters

and when you take the square root of square meters you're going to get the answer in meters

for those of you who want to see how the units work in that formula now let's go ahead and

plug in these values into a calculator so we can get our final answer so let's plug in the square root of nine

times ten to the nine times eight hundred times ten to the negative nine

times nine hundred times ten to the minus nine divided by fifteen so you should get this answer .02078

meters now let's go ahead and convert that to centimeters

because we want the answer to be in centimeters now remember one meter is equivalent to

100 centimeters so you want to set up the conversion in such a way that the unit meters cancel

so we're going to multiply .02078 by a hundred or simply move the decimal two units to

the right so we're going to get 2.078

centimeters so that's the distance between

the two point charges number three a force of 500 newtons exists between

two identical point charges separated by a distance of 40

centimeters calculate the magnitude of the two point charges so let's draw a picture

so here are the two point charges they're identical they can both be positive or both be negative now because

they're identical we don't have to describe them as q1 and q2

we can simply refer to them as q because they're the same

as the problem mentions they are identical now we know the distance between

the two point charges and so that is our r value which is 40 centimeters

now we want to convert that to meters so to quickly convert centimeters to meters simply divide by 100 or move the

decimal two units to the left so 40 centimeters is equivalent to 0.40 or 0.4 meters

now we have the magnitude of the force that repels these two charges away from each

other that's 500 newtons how can we calculate q well let's begin by writing the formula

f is equal to k now typically we would write q1 and q2

but we're going to replace q1 with q and q2 with q so

q times q is q squared now what we need to do is we need to isolate q squared

so i'm going to multiply both sides by r squared over k

the reason why i want to multiply both sides of the equation by r squared over k

is so that on the right we could cancel r squared and we can also cancel k so what we have left over on the right

side which is now on the left side because i reversed it

is q squared q squared is equal to f times r squared over k

now to get q we need to take the square root of both sides so for this particular problem

because the two charges are identical we can calculate q using this formula it's going to be equal to the force

times the square of the distance divided by k so the force is 500 newtons

r is point four meters and then k

is nine times ten to the nine with the units newtons

times square meters over square coulombs so we can see the unit newtons will cancel

and the unit square meters will cancel and so we're going to be left behind with square coulombs but when we take

the square root of that the unit will just be in columns so let's go ahead and plug this in

so the square root of 500 times 0.4 squared divided by 9 times 10 to the 9 all

within the square root symbol so you should get 9.428

times 10 to the negative 5 coulombs now to get the answer

in micro coulombs take this value and divide it by 10 to the negative 6 or

1 times 10 to the minus 6. so if you do that you should get 94.28 micro coulombs

so that is the magnitude of the two identical charges now let's move on to number four

how many electrons represent a charge of negative seventy microclones what we can do is we can convert this

to the number of electrons first let's convert it to coulombs we know one micro coulomb

is one times ten to negative six coulombs so we can cancel the unit micro coulombs

now we have the conversion factor between number of electrons and electric charge

keep in mind one electron has an electric charge of negative 1.6 times 10 to the negative 19 coulombs

it's really negative 1.602 but if you round it to negative 1.6 your answer will still be accurate

so let's put coulombs on the bottom and then one electron on top so now we can cancel the unit coulombs

so it's going to be negative 70 times 1 times 10 to the negative 6 divided by negative 1.6

times 10 to the negative 19. so the answer is 4.375

times 10 to the 14 electrons so that's how many electrons are represented by a charge of negative 70

micro coulombs now what about number five what is the electric charge in micro

coulombs of five times 10 to the 14 protons so for this problem we're going

backwards we're converting number of protons to electric charge

so let's start with what we're given 5 times 10 to the 14 protons now let's convert protons to charge we

know that one proton has a charge of positive 1.6

times 10 to the 19 coulombs and we know that one microclone is one times 10 to the negative 6

coulombs so the number of protons cancels and the charging coulombs cancel

so we get the charge in micro coulombs so it's going to be oh by the way this is negative 19.

so it's 5 times 10 to the 14 times 1.6 times 10 to the negative 19 and then take that result divided by 1

times 10 to the negative 6 and you should get 80.

so the charge that it represents is positive

80 micro coulombs so that's the answer for number five

number six a metal sphere has 4.3 times 10 to the 21 protons

and 6.8 times 10 to the 21 electrons what is the net electric charge on this metal sphere

in order to determine the net electric charge we need to know

if we have more protons than electrons or vice versa in this case we have more electrons

what we really need to do is determine the excess number of electrons and convert that to electric charge

but there's a simple formula that you could use to get this answer quickly so the net charge q

let me draw that better it's going to be the difference between the number of protons and electrons

so if we get more protons than electrons this number will be positive if the number of electrons exceeds the

protons this number will be negative so it's going to be p minus e times the charge of a proton which is

1.6 times 10 to the negative 19 coulombs so for this problem all we need to do is

plug in these values so we have 4.3 times 10 to the 21 protons

and we have 6.8 times 10 to the 21 electrons so 4.3 minus 6.8

that's a difference of negative 2.5 so because it's negative that tells us that the overall charge will be negative

and so that means that we have more electrons and protons so negative 2.5 times 10 to the 21

times 1.6 times 10 to the negative 19. this is equal to negative 400 coulombs

so that is the net electric charge on this particular metal sphere a 100 micro coulomb charge is placed at

the origin a negative 50 micro cooling charge is placed at x equals 2 meters

relative to the origin and a 200 micro coulomb charge is placed at x equals negative 4 meters

what is the net electric force acting on the 100 microcom charge so if you want to try this problem feel

free and uh pause the video to see if you can get the right answer so go ahead and work on this problem

now the first thing we need to do is draw a picture so we have a positive charge

placed at the origin and that's the 100 microclimate charge and then at a position of 2 meters

relative to the origin we have a negative charge with a magnitude of negative 15 micro

coulombs and then on the left side at negative four meters

we have a positive charge with a magnitude of 200 micro coulombs

so the distance between these two charges is four meters and between the two charges on the right

it's two meters let's call this q1 q2

and q3 so how can we use this information to determine the net electric force

acting on the middle charge first we need to identify all the forces acting on the middle charge

so q2 feels a force of attraction with q3 q2 is attracted to q3 so there's going to be a force that's

going to accelerate q2 towards q3 and we're gonna call that force

f two comma three so this is the force on two

due to charge three now q2 with its positive charge is repelled by q1

so q2 wants to move away from q1 so there's another force acting on q2 and that's going to be f2 comma 1

that is the force acting on charge two due to charge one

and so the net force is going to be the sum of these two forces anytime you have two

vectors that are parallel to each other to find the sum of those vectors you can simply add them

so the net force is going to be f 2 on 1 i mean f 1 and 2 rather

plus f 3 onto or the force exerted onto by a

charge 3. so we just got to add those two values so first we need to calculate each one

so let's calculate f 2 comma 1 so it's going to be k times q 1 times q2

divided by r squared where r is the distance between q1 and q2

so k is nine times ten to the nine q1 is 200 micro coulombs or 200 times 10 to the minus 6

and q2 is 100 times 10 to the negative 6

divided by r squared or 4 squared so go ahead and type that in so the force exerted on charge two

biocharge one is 11.25 newtons so let's save that answer we're going to

use it later now we need to calculate the force exerted on charge 2

by charge 3. so that's going to be k q 2 times q 3

over r squared so k is a constant it's not going to change

it's 9 times 10 to 9 q 2 is 100

times 10 to the minus six and q3 is 50

times 10 to the negative six don't worry about the negative sign these two forces are positive because

they're both directed along the positive x direction so you can adjust the sign based on the direction of the force

now r the distance between q 2 and q 3 is 2 meters so we're going to divide it by 2 squared

it turns out that f 2 comma 3 is equal to the same number

11.25 newtons purely by coincidence so now we can calculate the net force

acting on charge two so as we said before the net force is simply the sum of these

two forces because they're going in the same direction so it's f two comma one plus

f two comma three and so that's gonna to be 11.25 plus 11.25

which adds up to 22.5 newtons and so that's the answer for part a

now let's move on to the next part what is the net electric force acting on the 200 micro coulomb charge

so that's on q1 so let's see what effect q2 has on q1 q2 repels q1

so q2 will cause q1 to go in this direction so we can call that

f one comma two that's the force exerted on charge one by charge two

now q1 is attracted to q3 because opposites attract so we have another force

which we'll call f1 comma three that's the force exerted on charge one as a result of charge three

now these two forces are competing with each other now this force is in a positive x

direction so we're going to give a positive number this force is in a negative direction

so the net force is going to be the sum of these two values if the net force is positive

that means that f103 is greater which is probably not because it's so far away if the net

force is negative which is probably going to be the case that means that

this force is greater so let's calculate the values of each of these two forces

now it's important to understand that f 1 comma 2 is equal to

f2 comma 1 in magnitude they have the same magnitude but the direction may not be the same so the sign might be

different so technically we should say that the absolute value of f one comma two

is equal to the absolute value of f two comma one because one might be negative and the other might be positive

so we don't need to calculate this value because we already have it here we do however need to calculate this

value because we don't have that yet so f1 comma 3 that's going to equal k q1 q3 divided by r squared

now k is nine times ten to the nine q one is two hundred

times ten to the minus six q three is fifty times ten to the minus six and

the distance between q1 and q3 is four plus two or six meters

so this is going to be six squared on the bottom and so the force is going to be very

small so i'm going to write it over here it's only 2.5 newtons

because the charges are so far away the magnitude of the force will be greatly reduced

so now we can calculate the net force it's going to be f1 comma 2 plus

f1 comma 3. now this force is negative because it's in a negative x direction

and it has the same magnitude as that one so it's going to be negative 11.25 and this force is positive it's directed

in the positive x direction so that's positive 2.5 and so the net force is negative 8.75

newtons along the negative x direction so that's the net force acting on charge

one