In this problem we want to calculate the electric field due to several charge linear distributions .a) Consider a thin rod of length L. The rod carries a uniform charge density, λ, and total charge Q. Calculate the electric field at a point p that is on the rod’s cylindrical axis of symmetry and a distance d away from the end of the rod. Do this by direct integration. How does your expression be have as L→∞?b) Repeat, with the rod’s charge density being λ=αx, whereαhas a numerical value of 0.25 (what are the dimensions and units of α?)?

Two very long wires run in the z direction, one at (1,0) cm, the other at (-1,0) cm. (a) If both wires are carrying a positive current of 0.6 A upwards in z, calculate the force each wire exerts on the other. Is it repulsive or attractive? (b) Do the same problem, but with the wire at (-1,0) cm carrying 0.6 A downwards. Hint: you shouldn’t have to do any more math for this part

Compute the equivalent focal length of the following thick lens combination:

Lens A: R1 = 16.28mm, R2 = -275.7mm, n = 1.6116 , d = 3.57mm

Distance: 1.89 mm

Lens B: R1 = -34.57mm, R2 = 15.82mm, n = 1.6053, d = 0.81mm

Distance: 3.25mm

Lens C: R1 = inf, R2 = 19.20mm, n = 1.5123, d = 2.17mm

In contact with,

Lens D: R1 = 19.20mm, R2=-24mm, n=1.6116, d=3.96mm

Three spheres are arranged in the xy plane as in the figure below. The first sphere, of mass

m₁ = 15.6 kg,

is located at the origin; the second sphere, of mass

m₂ = 4.47 kg

is located at (−6.00, 0.00) m; and the third sphere, of mass

m₃ = 2.00 kg

is located at (0.00, 5.00) m.Assuming an isolated system, what is the net gravitational force on the sphere located at the origin?

A roller-coaster car speeds down a hill past point A where R1 = 11.6 m and then rolls up a hill past point B where R2 = 15.2 m, as shown below.

(a) The car has a speed of 20.0 m/s at point A. if the track exerts a normal force on the car of 2.08 104 N at this point, what is the mass of the car?

(b) What is the maximum speed the car can have at point B for the gravitational force to hold it on the track?

Discuss how Maxwell's predictions were confirmed by the experiments carried out by Hertz. Please use own words at least 2 paragraphs.

A thick lens has an index of refraction of 1.60. It is 4-cm thick and is surrounded by air. The radius of curvature of the front surface is 5 cm, and the radius of curvature of the back surface is 12 cm. A 1-cm tall object is placed to the left of the first surface. At what angle above the optic axis should a ray strike the front side of the lens 0.5 cm above the optical axis so that it emerges from the back side of the lens traveling parallel to the optical axis? Use matrix techniques.

A proton with an initial velocity of 3.9 106 m/s enters a parallel-plate capacitor at an angle of 46° through a small hole in the bottom plate as shown in the figure below. If the plates are separated by a distance
d = 2.1 cm,
what is the magnitude of the minimum electric field to prevent the proton from hitting the top plate?
___ N/C

Compare your answer with the results obtained if the proton was replaced with an electron and choose an answer:

A. The magnitude of the electric field needed for a proton is larger than what is needed for an electron.

B. The magnitude of the electric field needed for a proton is smaller than what is needed for an electron.   

C. The magnitude of the electric field needed for a proton is the same as what is needed for an electron.

Compare and contrast a simple pendulum with a mass/spring pendulum. Be sure to address velocity, acceleration, and energy conversions.

In the shot put, a heavy lead weight—the "shot"—is given an initial velocity, starting from an initial elevation approximately equal to the shot putter's height, say, 2.00 m. If v0 = 7.05 m/s, find the horizontal distance traveled by the shot for the following initial angles above the horizontal.

theta 0 =?

theta 40.0 = ?

theta 45.0 = ?

In 1610, Galileo used his telescope to discover four prominent moons around Jupiter. Their mean orbital radii a and periods T are as follows: (a) Io has a mean orbital radius of 4.22 x 108 m and a period of 1.77 days. Find the mass of Jupiter from this information. (b) Europa has a mean orbital radius of 6.71 x 108 m and a period of 3.55 days. Find the mass of Jupiter from this information. (c) Ganymede has a mean orbital radius of 10.7 x 108 m and a period of 7.16 days. Find the mass of Jupiter from this information. (d) Callisto has a mean orbital radius of 18.8 x 108 m and a period of 16.7 days. Find the mass of Jupiter from this information.

An object is launched at a velocity of 20 m/s in a direction making an angle of 25 degrees upward with the horizontal.

a) What is the maximum height reached by the object?

b) What is the total flight time (between launch and touching the ground) of the object?

c) What is the horizontal range (maximum x above ground) of the object?

d) What is the magnitude of the horizontal component of the velocity of the object just before it hits the ground?

e) What is the magnitude of the vertical component of the velocity of the object just before it hits the ground?

f) What is the magnitude of the total velocity of the object just before it hits the ground?

g) What is the angle with which the object hits the ground?

Problem 5. A non-conductive solid sphere has a volumetric charge distribution given by ρ (r) = (β / r) sin (πr / 2R). Where R is the radius of the sphere and β a positive constant. Find the total charge contained in the spherical volume and the electric field in regions r> R and r <R. Show that the two expressions for the electric field are equal in r = R.

The Nardo ring is a circular test track for cars. It has a circumference of 12.5 km . Cars travel around the track at a constant speed of 100 km/h. A car starts at the easternmost point of the ring and drives for 7.5 minutes at this speed. Part A What distance, in km, does the car travel? Express your answer in kilometers. Part B What is the magnitude of the car's displacement, in km, from its initial position? Express your answer in kilometers. Part C What is the speed of the car in m/s? Express your answer in meters per second.