Calculate the divergence of the given field at P(V2,315°,2): D %3DXze2 (г а, + хz а, + ха,)
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Question 3
Calculate the divergence of the given field at P(V2,315 °, 2):
D = xze²» (z a, + xz a, + x a,)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0148e06b-76a5-4769-be59-68e8a2a66854%2F4fc663a4-7886-4f94-9572-c8538e893358%2Ftcowpya_processed.png&w=3840&q=75)
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- An infinite cylinder of radius R has a charge density given by p(r) = ar", where r is the perpendicular distance from the axis of the cylinder, and a is a constant. Show that the electric field for r > R given by aR E(r) 7€or satisfies V ·Ē = 0 for r > R. Explain briefly why this condition must be obeyed.The co-energy of a device is given as Wm= %3D What is the flux linkage A when i = 0.84 A and z = 0.29 cm?. A[Wb turns]= number (rtol=0.001, atol=1e-08) What is the force of electric origin f"(i, x) for these conditions? f°IN]= number (rtol=0.01, atol=1e-08) What is the energy Wm for the same conditions?. HINT: Wm+ Wm = Xi Wm) = number (rtol=0.001, atol=1e-08)Problem Using the method of integration, what is the electric field of a uniformly charged thin circular plate (with radius R and total charge Q) at xo distance from its center? (Consider that the surface of the plate lies in the yz plane) Solution A perfect approach to this is to first obtain the E-field produced by an infinitesimal charge component of the charge Q. There will be several approaches to do this, but the most familiar to us is to obtain a very small shape that could easily represent our circular plane. That shape would be a ring. So for a ring whose charge is q, we recall that the electric field it produces at distance x0 is given by E = (1/ )(xOq)/( 2472) Since, the actual ring (whose charge is dq) we will be dealing with is an infinitesimal part of the circular plane, then, its infinitesimal electric field contribution is expressed as = (1/ )(x0 24 We wish to obtain the complete electric field contribution from the above equation, so we integrate it from 0 to R to…
- A 2D annulus (thick ring) has an inner radius Ri and outer radius Ro, and charge Q non-uniformlydistributed over its surface. The 2D charge density varies with radius r by η(r)=Cr 4 for Ri ≤ r ≤ Roand η=0 for all other r. C is a constant. Answer the following in terms of the variables given above. Note: Gauss' Law will not be useful here.a) Find an expression for C such that the total charge of the annulus is Q. Include the SI units for C next to your answer.Do the units make sense?b) Draw a clear picture and use it to set up the integral to calculate the E-field at a point on the axis of the annulus (this axis is perpendicular to the plane of the annulus) a distance z from the center of the annulus. *** You must complete all the steps short of computing the integral (i.e. your eventual answer must be an integral with only ONE variable of integration and all other variables constant.) Show that your answer has the correct SI units for the electric field.An infinitely long insulating cylinder of radius R has a volume charge density that varies with the radius as given by the following expression where pa, and are positive consta axis of the cylinder p-p(0-1) Use Gauss's law to determine the magnitude of the electric field at the following radial distances. (Use the following as necessary: Po, b, c, and R.) (a) R EM Po x 2R (-36) 200 xIn Millikan’s experiment, an oil drop of radius 1.64 mm and density 0.851 g/cm3 is suspended in chamber C when a downward electric field of 1.92 * 10^5 N/C is applied. Find the charge on the drop, in terms of e.
- A cylinder of radius a and height L is centered about the z -axis and has a uniform polarization alongits axis, P = P_0 z-hat. (P is a vector in the z direction).Find the electric field and electric displacement everywhere on its axis.Problem An insulating sphere with radius R contains a total non-uniform charge (i.e. Hydrogen atom) Q such that its volume charge density is 38 312 where Bis a constant and ris the distance from the center of the sphere. What is electric field at any point inside the sphere? Solution To find the electric field inside the non-uniformly charged sphere, we may apply integration method or the Gauss's Law method. Here, let us use the Gauss's law which is expressed as We will choose a symmetric Gaussian surface, which is the surface of a sphere, then evaluate the dot product to obcain (Equation 1) The issue however is how much charge does the Gaussian surface encoses? Since, our sphere is an insulating material. charges will get distributed non-uniformly within the volume of the object. So, we look into the definition of volume charge density to find the enclosed charge. So, we have dq p= dv Based on the given problem, we can also say that dgenc p= 38 dVProblem An electric charge Q is distributed uniformly along a thick and enormously long conducting wire with radius R and length L. Using Gauss's law, what is the electric field at distance r perpendicular to the wire? (Consider the cases inside and outside the wire) Solution To find the electric field inside at r distance from the wire we will use the Gauss's law which is expressed as We will choose a symmetric Gaussian surface, which is the surface a cylinder excluding its ends, then evaluate the dot product to obtain A = (Equation 1) Case 1: Inside the wire Since, r falls inside the wire, then all the enclosed charge must be: denc On the other hand, the Gaussian surface inside the wire is given by A = Using Equation 1, the electric field in simplified form is E =
- GggOA Solid insolatina Cylinder with positive uniform volume Charge density tP + radiusa, is surrounded beu a thin-walled insulativa Culindrical sell aith negative uniform sursace enarge densi ty-o and radius b. The two culinders ave coaxial (Ghave the Same central asis) + both have a lena th L. Use the Quantitiec given t a cu livdrical Gaus Sian surface. tp Dwrite Durite Qurite an expressi on Sor the Charge enclosed within the region arrrb an expression for the chavge enclosed within region rra expression S for the electric field within the tegion rra an write Durite an expression for the electric field within the region a Trb exoression Sor the Charge enclose d within the region r>b