Calculate the electric field at height h above the center of a square plate of size 2a×2a with uniform surface charge density η (both direction and magnitude). Verify that in the limit of large a the result agrees with the field of an infinite uniformly charged plane.
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- Sections AB and CD of a thin non-conducting ring of radius R are uniformly (with constant linear density) charged with charge + q and −q, respectively. The points ABCD form the vertices of the square. Find the electric field in the center of the ring.An insulating solid sphere of radius a has a uniform bulk density ρ and a total positive charge Q. Calculate the magnitude of the electric field at a point outside the sphere.A spherical insulator with radius R is centered at the origin. The volume charge density ρ of the insulator is non-uniform and varies with position according to ρ(r) =ar, where a is a constant. Derive an expression for the electric field E(r) for all points outside the sphere (r > R). Hint: finding the total charge of the sphere will require an integral.
- Positive charge Q is distributed uniformly along the x-axis from x=0 to x=a. A positive point charge q is located on the positive x-axis at x=a+r, a distance r to the right of the end of Q. Calculate the x-component of the electric field produced by the charge distribution Q at points on the positive x-axis where x>a. Express your answer in terms of the variables Q, a, r, and and appropriate constants. Calculate the magnitude of the force that the charge distribution Q exerts on q. Express your answer in terms of the variables Q, q, a, r, and appropriate constants. Calculate the direction of the force that the charge distribution Q exerts on q.Positive charge Q is distributed uniformly along the x-axis from x=0 to x=a. A positive point charge q is located on the positive x-axis at x=a+r, a distance r to the right of the end of Q. Calculate the x-component of the electric field produced by the charge distribution Q at points on the positive x-axis where x>a. Express your answer in terms of the variables Q, a, r, and and appropriate constants. Calculate the y-component of the electric field produced by the charge distribution Q at points on the positive x-axis where x>a. Express your answer in terms of the variables Q, a, x, and appropriate constants. Calculate the magnitude of the force that the charge distribution Q exerts on q. Express your answer in terms of the variables Q, q, a, r, and appropriate constants. Calculate the direction of the force that the charge distribution Q exerts on q.Find the electric field at the origin of the x,y-plane for charge distributions (a) and (b), see the figures shown below. The field is produced (a) by a thin half-circle with a radius of 12.5 cm and the linear charge density K = 18.5 pC/cm and (b) by a thin quarter-circle with the same radius and the linear charge density K = -18.5 pC/cm. K>0 y 0 (a) For the charge distribution (a): The x-component of Ea, Ea,x The y-component of Ea, Ea,y ² For the charge distribution (b): The x-component of Eb, Eb.x = = The y-component of Eb, Eb,y = K<0 Units Select an answer ✓ Units Select an answer ✓ Units Select an answer Units Select an answer 0 (b)
- Find the electric field at the origin of the x,y-plane for charge distributions (a) and (b), see the figures shown below. The field is produced (a) by a thin half-circle with a radius of 15 cm and the linear charge density K-10 pc/cm and (b) by a thin quarter-circle with the same radius and the linear charge density K = -10 pc/cm. K>0 (a) For the charge distribution (a): The x-component of Ea. Ea,x= The y-component of Ea, Ea,y= For the charge distribution (b): The x-component of Eb, Eb,x- The y-component of Eb, Eb,y = Units N/C Units N/C Units N/C Units N/C K<0 (b)Charge of a uniform density (11 pC/m?) is distributed over the entire xy plane. A charge of uniform density (6 pC/m2) is distributed over the parallel plane defined by z = 2.0 m. Determine the magnitude of the electric field for any point with z = 3.0 m.A point charge Q sits at the center of a hollow conducting sphere with a concentric surfaces of inside radius a and outside radius b. If the conductor is given a net charge q, then what are the surface charge densities σ (Q/A) at both surfaces of the conductor, at radius a and radius b?
- An infinitely long cylinder in free space is concentric with the z-axis and has radius a. The net charge density p in this cylinder is given in cylindrical coordinates by, 1 a² +r² where A is a constant. (a) Show that the total charge per unit length, λ in the cylinder is λ = πA ln 2. p(r) = A- Hint: you may find the following integral useful. 1 2 J for r a) and inside the cylinder (r< a). (d) The cylinder is composed of a material in which the polarisation P is given by P = P₁² in (1 +5²) e₁₁ er, r where Po is a constant. Determine the bound charge density pb in the cylinder. Hence, or otherwise, determine a relation between A and Po such that the free charge density of in the cylinder vanishes.An infinite line charge has constant charge-per-unit-length λ. Surrounding the line charge is a cylindrical shell of radius R, and carrying a constant charge-per-unit area σ. Given λ, what must σ be in order to get 0 electric field for all points outside the cylindrical shell? For that σ, what is the electrical field in between the line of charge and shell?An infinite plane slab, of thickness 2d, carries a uniform volume charge density (rho). Find the electric field, as a function of y, where y = 0 at the center. Plot E versus y, calling E positive when it points in the +y direction and negative when it points in the −y direction.