2.  The gravitational force between two masses, m1 & m2 is proportional to the product of the masses and inversely proportional to the square of the distance between them. F = G m 1 m 2 d 2

3. F = G m 1 m 2 d 2  G = the universal gravitational constant  m = mass  d = distance between the objects

4.  Electrical force between two objects/charges has a similar inverse- square relationship with distance.  Coulomb’s Law - for charged particles/objects, the force between the charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

5.  Coulomb’s Law gives you the magnitude of the force between the objects/charge. F = k q 1 q 2 d 2 d = distance between the charged particles q 1 = the quantity of charge of one particle q 2 = quantity of charge of the other particle k = the proportionality constant = 9.0 x 10 9 Nm 2 /C 2 The SI unit of charge is the coulomb (C). 1C = 6.24 x 10 18 electrons

6.  Remember that force is a vector, so when more than one charge exerts a force on another charge, the net force on that charge is the vector sum of the individual forces. Remember, too, that charges of the same sign exert repulsive forces on one another, while charges of opposite sign attract.

7.  What is the attractive electric force between a hydrogen atom’s proton and electron?  q p = + 1.6 x 10 -19 C  q e = - 1.6 x 10 -19 C  r = 5.3 x 10 -11 m (atomic radius)  k C = 8.99 x 10 9 N-m 2 /C 2

8. F elect = 8.99 x 10 9 N.m 2 /C 2 (1.6 x 10 -19 C) (1.6 x 10 -19 C) (5.3 x 10 -11 m) 2 F elect = 8.19 x 10 -8 N, attractive

9. One charge of 2.0 C is 1.5 m away from a – 3.0 C charge. Determine the force they exert on each other.

10. Fe= k q1 q2 r2 Fe= (8.99e 9 Nm 2 /C 2 )( 2.0 C)(−3.0 C) (1.52 m) 2 Fe= −2.4 e 10 N  The negative sign just means that one charge is positive, the other is negative, so there is an attractive force between them.

11. Things get a bit more interesting when you start to consider questions that have more than two charges.  You will almost always deal with three charges in these linear problems.  In the following example you have three charges lined up and are asked to calculate the net force acting on one of them.  Do one step at a time, and then combine the answers at the end.

12.  Example 2 : The following three charges are arranged as shown. Determine the net force acting on the charge on the far right (q3 = charge 3). q1= 1.5e -7 C q2 = - 2.3 e -7 C q3 =−3.5e -4 C 1.4 m 1.7 m

13.  Calculate the force between one pair of charges, then the next pair of charges, and so on until you have calculated all the possible combinations for that particular question.  Remember, if you've calculated the force of q1 on q2, then you also know the force of q2 on q1... they're the same!

14.  It does NOT matter that there is another charge in between these two… ignore it!  It will not effect the calculations that we are doing for these two. Notice that the total distance between charge 1 and 3 is 3.1 m, since we need to add 1.4 m and 1.7 m.

15. F 13 = k q1q3 r2 F 13 = (8.99e 9 Nm 2 /C 2 )( 1.5e -7 C )(−3.5e -4 C) (3.12 m) 2 F 13 =−4.9e -2 N

16.  The negative sign just tells us the charges are opposite, so the force is attractive. Charge 1 is pulling charge 3 to the left, and vice versa.  Do not automatically treat a negative answer as meaning “to the left” in this formula!!! Since all we care about is what is happening to charge 3,  All we really need to know from this is that, charge 3 feels a pull towards the left of 4.9e -2 N.

17. F 23 = k q2 q3 r 2 F 23 =8.99e 9 Nm 2 /C 2 )(−2.3e -7 C)(−3.5e -4 C) (1.72 m) 2 F 23 = + 2.5e -1 N  The positive sign tells you that the charges are either both negative or both positive, so the force is repulsive. We know that charge 2 is pushing charge 3 to the right with a force of 2.5e -1 N.

18.  Force acting on charge 3 is Fnet 3 = F 13 + F 23 F 13 = −4.9e -2 N F 23 = +2.5e -1 N Fnet 3 = −4.9e -2 N + 2.5e -1 N Fnet 3 = 2.01e -1 N

19.  Calculate the magnitude and determine the direction of the electric force between q 1 and q 2 and those between q 1 and q 3 per the diagram below: q1q1 q2q2 q3q3 +4 μ C +2 μ C - 3 μ C 40 cm 60 cm

20.  Calculate the magnitude and direction of F elect. on charge q 1 in the example below  K c = 8.99 x 10 9 N-m 2 /C 2  F elect. = k c q 1 q 2 /r 2 3.0 cm 2.0 cm q1q1 q2q2 q3q3 + 6.0  C+ 1.5  C- 2.0  C