1) Three charged particles are arranged as shown below. Using the coordinate system shown the particle with charge -2q is at position (-a, 0) and the two particles With charge +q are at positions (0,a) and (0,-a) respectively,

a) In temis of x, a and q derive a formula for the electric field vector for points along the positive x axis, Express your answer using unit vector notation.

b) Where on the positive x axis (including x = 0) will the field magnitude be the greatest° in which direction will the field vector point when the magnitude is at its maximum? Explain how you know.

c) imagine that the particle with charge -2q is moved to the origin of the coordinate system and the two positively charged particles remain in place.

Derive a formula kg the electric field vector for points along the positive y axis for y>a in terns of y a and q.

Use unit vector notation.

2) The polar molecule in the diagram below has a positive charge of + 19e at one end and a negative charge of -19e at the other. It is immersed in a nonuniform electric field (and aligned with the field direction). The magnitude of the field is 4.6 x 10⁷ N/C at the negative end of the molecule and dmps: by 3% at the positive end.

a) Determine the net force that acts on the molecule (calculate the maiznititide and describe the direction).

b) What would the net force on the molecule be if the field were uniform with a magnitude (everywhere) of 4.6 x 10⁷ N/C?

c) Water is a polar molecule. Imagine a stream of water falling through a region of space where an electric field exists. How could this experiment be used to determine if the electric field is uniform or non uniform? Explain your reasoning and describe how you could use your observations to reach a conclusion.

3) In the device below a 500 V difference of potential is maintained between two conducting surfaces: one is roughly a hemisphere and the other a flat plate. The space between the conductors is a vacuum and the electric field in this space is non uniform. Several equipotential surfaces are shown. An electron crosses the 300 V equipotential surface heading in the general direction of the metal plate with a speed of 1.45 x 10⁷ m/s.

a) Determine the electron's kinetic energy as it crosses the 300 V surface. How much kinetic energy will the particle have when it reaches the 100 V surface? Also determine its speed at this point.

b) How much kinetic energy will the particle have as it reaches the metal plate (assuming it does)?

c) Does the electric field (generally) point in the direction of the flat plate or the hemisphere? How can you tell? Discuss.

4) The electric field in a region of space is directed along the x axis and can be described by the vector E = ρ/ε_o xi^{^} where ρ represents the volume density of charge in the region (for your purposes it's simply a constant), and ε_o = 1/4Πk_e.

a) As discussed in class if V(x) is the electric potential that describes the field then E == ρ/ε_o xˆi = - dV/dx ˆi. Find/derive a function V(x) that describes this field. That is, what function V(x) has derivative -ρ/ε_ox ? Note that there are "a million" of them as your result should include an arbitrary integration constant C. (see the examples we've done in class.)

The field described in part a (and the potential your derived) could describe the state to the right of (and including the surface) of the metal plate below. Assume there is positive charge on the plate and a uniform distribution of positive charge throughout the volume. The surface of the plate is at the origin.

b) With this in mind (and using your V(x)) what should be the value of C if the potential on the surface of the plate is 0 Volts? What should be the value of C if the potential is 100 V. Explain/show how you know.

c) If ρ = 4.4 x10^-9 C m³ find the value of the potential 10 cm to the right of the metal plate using each of the C's you just found. Will the value of ΔV = V (0.1m) - V (0) depend on the value of C? What about the gain or loss of kinetic energy by a charged particle that moves from the plate to the 10 cm mark? Will it depend on C? Explain/show how you know.