A rod of length L lies along the x-axis with its left end at the origin. The rod has a non-uniform charge density λ = ax, where a is a positive constant. (a) Express the total charge on the rod in terms of a and Z. (b) Calculate the electric field at point P, shown in the Figure. Take the limitd > L. What does the electric field look like in this limit? Is this what you expect? Explain. Hint: the following integral may be useful: a x dx = +In(x+a) (x+a)² x+a x² In(1+x)=x-+ x 2 (for small .x)
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- A non-uniform thin rod is bent into an arc of radius R. The linear charge density λ of the rod depends on θ and is given by λ = λ0 / cos θ where λ0 is a positive constant. The arc extends from θ = π/4 to θ = 3π/4 as shown in the image. a) Sketch the direction of the resultant electric field at the origin. b) Calculate the magnitude of the electric field E .Please answer the subparts A&B with the step solution. Im needed in 30 minutes thank uConsider a nonconducting sphere and a concentric nonconducting spherical shell, shown in the figure. The inner sphere has non uniform charge density ρ1 = ρ0 / r, where ρ0 is a constant, in units of C/m2 and radius a. The outer spherical shell has uniform charge density ρ2 , inner radius b, and outer radius c. Find the magnitude of the electric field inside the inner sphere, for r < a, in terms of the given variables, as needed.
- In the figure a nonconducting rod of length L = 8.26 cm has charge -q = -4.55 fC uniformly distributed along its length. (a) What is the linear charge density of the rod? What are the (b) magnitude and (c) direction (positive angle relative to the positive direction of the x axis) of the electric field produced at point P, at distance a = 14.1 cm from the rod? What is the electric field magnitude produced at distance a = 77 m by (d) the rod and (e) a particle of charge -q = -4.55 fC that replaces the rod?What is the electric field at the center of a sphere of radius R whose volume is uniformly filled with charge (total charge Q)? For this sphere, how does the electric field at a distance r1=R/4 from the center compare to the electric field at a distance r2=3R/4 from the center? Give your answer in the form E(r1)=(some constant)E(r2). For this sphere, how does the electric field at a distance r1=R from the center compare to the electric field at a distance r2=2R from the center? Give your answer in the form E(r1)=(some constant)E(r2).Which of the following charge distributions will generate a uniform electric field at the specified distance? Select all that apply. a) A distance r from the center of a long, thin line of uniform charge with length L where r << L. b)A distance r away from a large disc of uniform charge (with radius R) where r << R (and not near the edge of the disc) c)A distance r away from a sheet of uniform charge (with dimensions a x a) where r << a (and not near the edge of the sheet). d)Any point between two infinitely large sheets of uniform charge, one with total charge +Q and the other with total charge -Q. e)A distance r from the center of a sphere of uniform charge (with radius R) where r >> R. f)Halfway between two infinitely large sheets of uniform charge, each with total charge -Q.
- a) A very long (almost infinitely long) cylindrical wire of radius R carries a uniform charge density Po. Find the line charge density. Find the electric field inside and outside the wire. b) If a long cylindrical cavity of radius b is created at a distance a in the wire maintaining same charge density as part (a) (see Fig. 2). Find the line charge density. Find the electric field inside the cylindrical cavity. 01 Fig. 2Here’s one more model of an electron. Here e is the magnitude of the charge of an electron, and R is a constant parameter characterizing the electron’s size: rho=-e (105/4pi)r2(R-r)2/R7 , rho = 0 for r >R. a. Use the problem's symmetry to evaluate the electric field at all points in space, it r >R and r<R. b. Sketch the electric field magnitude versus r for all values of r. d. evaluate the electric potential at all points in space and sketch its magnitude as a function of r.A thin glass rod is bent into a semicircle of radius r. A charge +Q is uniformly distributed along the upper half, and a charge –Q is uniformly distributed along the lower half, as shown in the figure. Find:1) The direction of the electric field at the center O of the semicircle as a function of Q and r. 2) The magnitude of the electric field at the center O of the semicircle as a function of Q and r.3) The force felt by a charge q0 = 2.0 x 10-7 C if this charge is placed at point O.
- Problem 14: Consider the semiarc of charge shown in the figure, that has an opening angle of θ0 = 35° and a charge density of λ = -23 μC/m. We are going to find the electric field at the point P, which is a constant distance R = 1.45 m from the semiarc. Use the coordinate system specified in the figure. Part (b) What is the x-component of the electric field at point P, in newtons per coulomb?You have a very (infinitely) long solid conducting cylinder with length L, base radius R, and total charge +Q. (a) Use Gauss’s law to find the electric field vector inside and outside the cylinder. Explain your reasoning. You can give a verbal description of the electric field vector direction. (b) Similar to how we found the electric field outside a conducting sphere to look like that of a point particle, what does the electric field outside the cylinder look like? (c) Would the electric field inside the cylinder stay the same if the cylinder was instead insulating and uniformly charged? Explain why or why not?Using Gauss's Law to calculate the electric field of a spherical object 2 Question 12: a) What is the volume charge density p = for a uniformly charged solid insulating sphere of radius R and with total charge Q? Sketch a graph of p as a function of radius, r; note the radius R is marked in the graph. R b) For some radius r R outside the uniformly charged solid sphere, how much charge is contained within a sphere of radius r?