Electrostatics

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4. Assume that an observer calculates electric field and potential by measuring the force on a test charge. This observer moves along the path described by the green arrow. Finally, assume the system is in vacuum. (a) Plot and justify |E| for a charged sphere-Q enclosed within an ungrounded spherical metal shell. (b) Draw all relevant field lines on the schematic (within reason). Hint: consider where the charges would be induced on the inner and outer surfaces of the shell.


To make a semiconductor conducting, it must have some carriers to flow the current. Answer following questions about carriers. (i) What are the different types of carriers that are possible in semiconductor lattice? (ii) What are the charges on different types of carriers? (iii)What are different methods by which we can increase the number of carriers in a semiconductor system? (Name two methods)


\text { (a) } V_{1}=10 r^{3} \sin 2 \phi \text { (b) } V_{2}=\left(2 / R^{2}\right) \cos \theta \sin \phi 3.60 Find the Laplacian of the following scalar functions:


Consider the following two mass-spring-damper system: The equations of motions for the system shown in Figure 1 are: m_{1} \bar{x}_{1}+c_{1} \dot{x}_{1}+k_{1} x_{1}-c_{1} \dot{x}_{2}-k_{1} x_{2}=f m_{2} \bar{x}_{2}+\left(c_{1}+c_{2}\right) \dot{x}_{2}+\left(k_{1}+k_{2}\right) x_{2}-c_{1} \dot{x}_{1}-k_{1} x_{1}=0 a) Implement the system of equations above in Simulink using the following parameters: m1 = 10; m2 = 100; c1 = 100;% I c2 = 1000; k1 = 1e4; k2 = 1e5; Define the model parameters in a separate .m file and use the ode45 Solver inside of Simulink. Make sure to decrease the maximum step size if the plots are not smooth. b) Simulate the response of the system assuming that f(t) is a step function of magnitude 5N. Plot the response of the systems (the two positions x1(t) and x2(t)) in two separate figures. c) Simulate the response of the system assuming that f (t) is a sinusoidal function:flt)=3 sin (10t). Plot the response of the systems (the two positions x1(t) and x,(t)) in two separate figures.


4. Given a parallel RLC circuit with wo = 10rad/s, Q = 1/2, and C = 1F, let the inputbe a current source i,(t) = u(t)cos2t connected in parallel. Determine the completeresponse for the initial conditions vc(0-) = 2V, and i¿(0¯) = 5A. Indicate clearlythe transient part, and the steady-state part. Take the inductor current as the response.


1. (Optional) Show that: \text { a. } \frac{d}{d t} e^{-\alpha t} \cos \omega_{d} t=-\omega_{0} e^{-\alpha t} \sin \left(\omega_{d} t+\phi\right) \text { b. } \frac{d}{d t} e^{-\alpha t} \sin \omega_{d} t=+\omega_{0} e^{-\alpha t} \cos \left(\omega_{d} t+\phi\right)


Insulator Conductor Semiconductor Intrinsic semiconductor Extrinsic semiconductor N-type semiconductor P-type semiconductor P-N Junction Forward biased Reverse biased Drift Current Diffusion Current Bipolar Junction Transistor (BJT) NPN PNP metal-oxide-semiconductor field-effect transistor (MOSFET) NMOS PMOS Diode-transistor (DTL) logic Resistor-transistor (RTL) logic Transistor-transistor (TTL) logic Emitter-coupled (ECL) logic Complementary MOS (CMOS) logic Bipolar CMOS (BICMOS) logic


3.46The scalar function V is given by \text { (a) Determine } \nabla V \text { in Cartesian coordinates. } V=\frac{2 z}{x^{2}+y^{2}} (b) Convert the result of part (a) from Cartesian to cylindrical coordinates. (c) Convert the expression for V into cylindrical coordinates and then determine VV in those coordinates. Compare the results of parts (b) and (c).


11: An Amperian loop surrounds a current as is shown on the left. Initially that current passes through the center of the loop. The current is then moved off center. Did the circulation of B around the loop change? In other words, did f B- ds change? Did the magnetic field on the loop change? Explain.


1. Assume that an observer calculates electric field and potential by measuring the force on a test charge. This observer moves along the path described by the green arrow. Finally, assume the system is in vacuum. (a) Plot and justify Q, E, V for a uniformly charged cylinder of P3D, height h, radius ro where h >> ro. The path goes through the center of the prism and is perpendicular to it. Let us assume V = 0 is defined at distance 1₁. (b) Suppose the observer is far enough away that z » h. Discuss whether it is appropriate to express E, V as if the charge were a point charge.


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