Derive the expression for the electrostatic field generated by a uniformly charged sphere (volumetric charge density p, dielectric constant so) of radius a, outside (r > a) and inside (r< a) the sphere. Show that the electric field is continuous at r= a.
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- An insulating sphere of radius R has a non-uniform charge density given byp = Ar 2 for r R. Find the electric field both inside and outside the sphere.A charge distribution creates the following electric field throughout all space: E(r, 0, q) = (3/r) (r hat) + 2 sin cos sin 0(theta hat) + sin cos p (phi hat). Given this electric field, calculate the charge density at location (r, 0, p) = (ab.c).The dome of a Van de Graaff generator receives a charge of 8.9 x 10-4 C. Find the strength of the electric field in the following situations. (Hint: Review properties of conductors in electrostatic equilibrium. Also, use the points on the surface are outside a spherically symmetric charge distribution; the total charge may be considered to be located at the center of the sphere.) (a) inside the dome N/C (b) at the surface of the dome, assuming it has a radius of 2.8 m N/C (c) 8.3 m from the center of the dome N/C
- A uniform charge distribution of total charge Q (Q>0) is located along the x-axis. The left end of the charge distribution is located at x=-(144/13)m and its right end is located at x=(25/13)m. Our goal is to calculate the electric field at point P which is located at (x,y)=(0,d). Use letters k for the Coulomb constant, Q for the total charge, d for the y-coordinate of point P, and x and y, for the cartesian coordinates in your responses. (a) Determine the expression for dq. (b) Determine the expression for r2, the magnitude of the vector that is directed from the element of charge dq to the location in which the field is to be determined. (c) Determine the expression for r̂ the unit vector that is directed from the element of charge dq to the location in which the field is to be determined (d) Write the expression for d due to the element of charge dq. (e) Let d=(60/13)m. Integrate the expression to determine the electric field at P. Perform this integration by hand. Avoid using…When you polarize a neutral dielectric, charge moves a bit, but the total charge remains zero. This fact should be reflected in the bound charges σ and ρ. Prove from σb = P. n and Pb = −√. P that the total bound charge vanishes.Ra1 +9 -9 Rea Consider two concentric spherical conductors, separated by an isolating material with (absolute) permittivity e. The two conductors have radius R1 and R2, they are put on a potential V and V2, which leads to a charge +q and –q sitting on them, respectively. By the problem's spherical symmetry, we see that the charge on each conductor is distributed uniformly, and that, in spherical coordinates, the electric field between the two conductors is of the form E(r) = -E(r) er. Determine the capacity C using the following steps: 1. Use Gauss's Law in integral form, with N a ball of radius r (R2 < r < R1), to find an expression for E(r) in terms of q. 2. Calculate AV = Vị – V2 using the formula - E•dr Δν and with C the black line segment indicated on the drawing (parallel with e,). 3. The capacity now follows from C = q/AV.
- A proton is held stationary at distance d, from an infinitely large conducting plane (sheet) of charge with uniform charge density -o. It is then released and moves straight towards the plane. Find the speed of the proton when it is at distance d, from the plane. The electric field of the plane is uniform everywhere in space and has the magnitude E = Ignore the gravitational force. Present your answer symbolically.Find the electric field vector anywhere in the plane of a dipole. Let the charge value on one charge be q. Let them be separated by d. Let the origin be in between them. And say they are each on the y axis.Find the electric field a distance z above the midpoint of an infinite line of charge that carries a uniform line charge density λ .