**Homework 2 – Physics 181, Spring Term 2021**

Please show your work in detail, without skipping steps and always write down the appropriate formula before you plug in numbers to calculate an answer. Do not use formulas we did not derive in lecture. Present clear sketches with properly projected forces in support of your symmetry arguments whenever applicable. This assignment covers lectures 3, 4 and 5, and focuses on the electric field.

Please upload your entire homework as a __single__ pdf file on Canvas under Assignments. Make sure to check that your __entire__ pdf has uploaded properly. No credit will be given to partially uploaded files. Problems 1, 2, 8 and 9 are for practice and don’t get submitted.

1. Three point charges lie along a circle of radius r at angles of 30^{o}, 150^{o}, and 270^{o} as shown in the figure below. Find the symbolic expression (in unit vector notation) for the resultant electric field at the center of the circle.

2. Find the symbolic expressions for the net electric field at points P and P^{’} in the figure below in unit vector notation. The charges located at A and B are +Q and –Q respectively. Make sure to bring your expression to simplest form. Keep the trig functions of the given angle in fraction form.

3. The figure below shows an arrangement of three charges along the x-axis: at the center, there is a negative charge -2Q flanked by a +Q charge on either side. The distance between each positive charge and the negative charge is *a*.

- Determine the electric field at a point A along the y-axis. Make sure to indicate the direction of the individual electric field vectors due to each charge and appropriate projections.
- If
*y>>a*, using a Taylor series approximation including terms of second order, work out how the electric field varies with y in this approximation.

4. An infinitely extended rod, lined up with the y-axis as seen in the figure below carries a positive uniform charge distribution, such that across a length L there is an amount of charge Q. Work out the components of the electric field due to this charge distribution at a point P on the x-axis in terms of Q and L and calculate the angle that the electric field makes with the horizontal.

5. A thin rod of length L, aligned horizontally with the x-axis, with one end at x=0 and the other at x=L, is carrying a non-uniform positive charge distribution with the linear density of charge given as µ(*x*)=*ax *, where *a* is a constant. The rod carries a total charge *Q* across its length.

- Determine the constant
*a*in terms of*Q*and*L*. - Determine the electric field vector due to this charge distribution at
*x=D*, in terms of*Q, L*and*D*. - Compare your answer with the case where µ=
*const*. and explain why it makes sense.

6. For the previous problem, what is the electric field in the event that D>>L? Hint: Notice what happens if you don’t carry the series expansion to second order!

7. Two point charges, *Q*_{1 }=−*Q*,*Q*_{2 }= 2*Q *, with Q>0, separated by a distance x, are located between two uniformly charged parallel plates. The plates are identical in size and shape. The surface charge density of the positively charged plate is three times as large as that of the negatively charged plate, i.e. σ_{1 }= 3σ,σ_{2 }=−σ, where σ>0 . Assume that the electric field produced by the charged plates is uniform, E. The known quantities are E, Q, and x.

- In a figure, represent the direction of the net electric field due to both plates and derive an expression for each surface charge density.
- Represent all forces on each particle, and work out a simplest form expression for the net force on each, in unit vector notation.
- If the net force on particle 1 is zero, what is the force on particle 2 and what are the surface charge densities on the plates?
- If the net force on particle 2 is zero, what is the force on particle 1?
- If E=45,000 N/C, calculate σ
_{1}and σ_{2}. Note that this is the only part where you are doing a calculation!

the mass of each sphere? Please disregard the too10*o *seen in the figure below.* ^{o}* 8. The identical small spheres shown in the figure below are charged to q=100nC as seen below. The external uniform electric field is 100,000N/C, θ=30 and L=50cm. What is

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