1. Chapter 21- Examples

2. Problem
Two free point charges +q and +4q are located a distance, L, apart. A third charge is placed so that the whole system is in equilibrium
Find the location, magnitude, and sign of the third charge

3. Step 1: Draw IT! +q
+4q L

4. Step 2: Think about it! +q +4q 3 1 2 L
There are 3 positions available for the new charge (marked in blue).
Which position do you think is the most likely position and more importantly why?
Positions 1 & 3 are untenable since by equilibrium, we mean the force on each particle is zero. If 1, for example, then there will be a repulsive force on +4q that cannot be overcome without creating a strong attractive force on +q

5. Step 3: Now a free body problem+q
+4q 2 L x F1 F2
Call the new charge, q0, and let it be distance x from +q
The free-body diagram shows relationship of the two forces, F1 and F2 on the new charge

6. Step 4: Use the other equilibrium points to derive more information2 L x F1
The free-body diagram shows relationship of the two forces, F1 and F4 on charge +q

7. Problem
An electron is projected with an initial speed of v0, 1.6 x 106 m/s in the uniform field between the parallel plates shown below. Assume that the plates are uniform and the field is directed vertically downward and the field outside the plates is zero. The electron enters the field at a point midway between the two plates
If the electron just misses the upper plate as it emerges from the field, find the magnitude of the electric field
2.0 cm
1.0 cm e-

8. Step 1: Draw It
Obviously, the electron is 0.5 cm below the upper plate when entering the field so, the initial position is (0,0.5) and the final position is (2.0, 0.0)
SI: (0,0.005) and (0.02, 0)
Also, v0 is actually in the x-direction so vinitial= (v0,0)
2.0 cm
1.0 cm e-

9. Step 2: Think about it!
This reminds me of a projectile problem except it is kinda upside down.
In this case, there is no acceleration in the x-direction only in the y and since F=ma=eE then a=eE/m
Where e=charge of electron and m is the mass of the electron.
Since I can never remember the trajectory equation, then I must solve for the time it takes the electron to traverse 2 cm and then plug that into the an equation relating acceleration to distance in the y-direction
2.0 cm
1.0 cm e-

10. Step 3: Equation Time
2.0 cm
1.0 cm e-

11. Problem
What is the magnitude and direction of the electric field at the center of the square if q= 1.0 x 10-8 C and distance, a, is 5 cm +q
-2q
+2q -q a

12. Field Lines Free body diagram of + test charge in center +q -2q +2q -q

13. Need break down into x and y components
First find the magnitudes of all the electric fields

14. For example, the field line from +2q
The line from +2q makes an angle of 1350 with the positive x-axis So
E2qx=E2q cos (1350)=-E2q cos (450)
E2qx=-0.707*E2q
E2qy=E2q sin (1350)=E2q sin (450)
E2qy=0.707*E2q
1350

15. And from -2q, (at 450)
E-2qx=0.707*E2q
E-2qy=0.707*E2q

16. And –q and +q -q makes angle of 2250 E-qx=-0.707*Eq E-qy=-0.707*Eq
Now we add the x and y components, respectively

17. Ex and Ey