Find the electric field E(x, y, z) for the electric potential given by V(x, y, z

Question

Find the electric field E(x, y, z) for the electric potential given by V(x, y, z) = yz + xz + 0.5z^2. (b) Evaluate the vector electric field E(x, y, z) at the position r = (1, -05.100)m. The density of charge on a thin rod is given the function lambda (x) = 0.01 x^4 in MKS units. (a) What is the total charge if the rod is 2 cm long and extends from x = 1.0 cm to x = +1.0 cm? b) Is there an electric field at the origin? (Do not calculate the field, just tell me why the answer either must be yes or must be no.) Some neutral hollow copper spheres with a radius of 2 cm have a mass of 3g. A total of 4 times 10^12 electrons are transferred from one sphere to another. If the electrostatic force of attraction between the spheres is equal to the weight of either one of the spheres, what is the separation between them? (a) A wire with a length of 10.0 m and a square cross-section (1.0mm times 1.0mm) is made of a material with a resistivity of 5.0 times 10^8 Ohm middot m What is its resistance? (b) If a potential difference of 1.00V is maintained between the ends of the wire, at what rate is heat generated in the wire (i.e. power consumption)? (c) Three of these wires (identical) are placed side-by-side (in parallel). What is the new power consumed assuming the same power supply is attached? (d) Five of the wires are now attached end-to-end (series). What is the new power consumed? (a) Draw for me a diagram of a battery in series with a capacitor and a resistor (values V_o, C and R respectively). (b) Set up Kirchhoff's loop rule for this circuit, which involves both q(t) and I(t). If you differentiate the entire equation, it can be re-expressed as a simple differential equation in terms of I(t) only. Find this diff. eq. (c) What is the standard solution for I(t) from this diff. eq.? (d) If R = 10.0 M Ohm, C = 1.00 mu F, V_o = 24V, and I_o = I(theta) = V_o/R, when does I = 2.00 mu A? (e) If I_o does not equal V_o/R, the solution is a bit more complicated, but it boils down to the standard solution shifted in time the left. What is I(5s) if I(0) = 2.00 mu A, using all the data from (d) above? A circular horizontal loop of one turn and radius 5 cm is laid on a table with its axis parallel to a vertical magnetic field of 0.6 T directed upward. By means one of some very powerful explosives, the area of the loop is suddenly reduced to essentially zero in 0.5 ms. (a) What emf is induced in the loop? (b) Looking down from above, is the induced current clockwise(CW) or counter-clockwise (CCW)? A monatomic gas expands from V_1 = 2 L to V_2 = 4 L while its pressure doubles uniformly from P_1 = 1 atm to P_2 = 2 atm. Then the pressure is reduced from P_2 back to P_1 while the volume continues to expand to V_3 = 6 L, again uniformly (a) If the gas is initially at STP, how many atoms are in the container? (b) How much work is done by the gas along this path? (c) What is the change in internal energy of the gas? (d) How much heat had to be supplied during this change? An iron plate is held against a stone wheel so that there is a sliding frictional force of 50 N acting between the two. The relative speed at which the two surfaces slide over each other is 40 m/s. (a) Calculate the rate at which mechanical energy is being dissipated in the form of thermal energy. (b) The plate and the wheel each have a mass 5.0 kg, and each receives 1/2 the thermal energy released. If the system is allowed to run for one minute, what is the final temperature of each? (Room temperature is 22 degree C, the specific heat of iron is 450 J/kg C degree, while that the is 750 J/kg C degree) (c) Based on this data, do you think that this metalworking task might be a hazardous source of serious skin burns? A transformer has N_p = 1000 turns in its primary and N_s = 25 turns in its secondary. The primary is connected to a 120 V, 60 Hz source. The secondary is connected to a 0.6 Ohm resistor. Assuming the transformer is 100% efficient (i.e. energy is conserved), calculate the current that flows in the primary. A mudded electric car is powered by sixty lithium-iron-phosphate batteries. At a freeway speed of 108 km/h, the average frictional force on the vehicle is only 200 N. (a) What is the power provided to the electric motor when the car travels at freeway speeds? (b) Each battery can store an energy of 0.1 kW.hr. Convert this to Joules, then find the total energy (in Joules) that can be delivered by the sixty batteries before they run dry. (e) How many minutes can the car nun before the batteries run out of power? (d) How many kilometers can the vehicle drive before it needs to be recharged? (e) The combination of batteries of delivers 24A-hr of charge. Convert this into Coulombs. (f) Using simple unit analysis, what is the output voltage of the array?

Explanation / Answer

For transformer

Np/Ns=Vp/Vs

1000/25=120/Vs

We got Vs=3 volts

Current in secondary =3/0.6=5 A

Again

Np/Ns=Is/Ip

1000/25=5/Ip

We got Ip=5/40=0.125 A or 125 mA

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