KTO Karatay University Engineering Faculty PHYS-‐102 Recitation Problem Set-‐2 PROBLEM SET-2 (Gauss’s Law) 1- An electric field with a magnitude of 3.50 kN/C is applied along the x axis. Calculate the electric flux through a rectangular plane 0.350 m wide and 0.700 m long assuming that (a) the plane is parallel to the yz plane; (b) the plane is parallel to the xy plane; (c) the plane contains the y axis, and its normal makes an angle of 40.0° with the x axis. 1 KTO Karatay University Engineering Faculty PHYS-‐102 Recitation Problem Set-‐2 2- Consider a closed triangular box resting within a horizontal electric field of magnitude E=7.80x104 N/C as shown in Figure. Calculate the electric flux through (a) the vertical rectangular surface, (b) the slanted surface, and (c) the entire surface of the box. 2 KTO Karatay University Engineering Faculty PHYS-‐102 Recitation Problem Set-‐2 3- A uniform electric field aî+bĵ intersects a surface of area A. What is the flux through this area if the surface lies (a) in the yz plane? (b) in the xz plane? (c) in the xy plane? 3 KTO Karatay University Engineering Faculty PHYS-‐102 Recitation Problem Set-‐2 4- Four closed surfaces, S1 through S4, together with the charges -2Q, Q and -Q are sketched in Figure. (The colored lines are the intersections of the surfaces with the page.) Find the electric flux through each surface. 4 KTO Karatay University Engineering Faculty PHYS-‐102 Recitation Problem Set-‐2 5- A positive point charge Q is located at the center of a cube of edge L. In addition, six other identical negative point charges q are positioned symmetrically around Q as shown in Figure. Determine the electric flux through one face of the cube. 5 KTO Karatay University Engineering Faculty PHYS-‐102 Recitation Problem Set-‐2 6- A particle with a charge of -60.0 nC is placed at the center of a nonconducting spherical shell of inner radius 20.0 cm and outer radius 25.0 cm. The spherical shell carries charge with a uniform density of -1.33 ︎C/m3. A proton moves in a circular orbit just outside the spherical shell. Calculate the speed of the proton. 6 KTO Karatay University Engineering Faculty PHYS-‐102 Recitation Problem Set-‐2 7- Consider a long cylindrical charge distribution of radius R with a uniform charge density ρ. Find the electric field at distance r from the axis where r<R. 7 KTO Karatay University Engineering Faculty PHYS-‐102 Recitation Problem Set-‐2 8- An insulating solid sphere of radius a has a uniform volume charge density and carries a total positive charge Q. A spherical gaussian surface of radius r, which shares a common center with the insulating sphere, is inflated starting from r=0. (a) Find an expression for the electric flux passing through the surface of the gaussian sphere as a function of r for r<a. (b) Find an expression for the electric flux for r>a. (c) Plot the flux versus r. 8 KTO Karatay University Engineering Faculty PHYS-‐102 Recitation Problem Set-‐2 9- A square plate of copper with 50.0 cm sides has no net charge and is placed in a region of uniform electric field of 80.0 kN/C directed perpendicularly to the plate. Find (a) the charge density of each face of the plate and (b) the total charge on each face. 9 KTO Karatay University Engineering Faculty PHYS-‐102 Recitation Problem Set-‐2 10- A long, straight wire is surrounded by a hollow metal cylinder whose axis coincides with that of the wire. The wire has a charge per unit length of λ, and the cylinder has a net charge per unit length of 2λ︎. From this information, use Gauss’s law to find (a) the charge per unit length on the inner and outer surfaces of the cylinder and (b) the electric field outside the cylinder, a distance r from the axis. 10 KTO Karatay University Engineering Faculty PHYS-‐102 Recitation Problem Set-‐2 11- A thin square conducting plate 50.0 cm on a side lies in the xy plane. A total charge of 4.00x108 C is placed on the plate. Find (a) the charge density on the plate, (b) the electric field just above the plate, and (c) the electric field just below the plate. You may assume that the charge density is uniform. 11 KTO Karatay University Engineering Faculty PHYS-‐102 Recitation Problem Set-‐2 12- A nonuniform electric field is given by the expression E=ayî+bzĵ+cx kˆ , where a, b, and c are constants. Determine the electric flux through a rectangular surface in the xy plane, extending from x=0 to x=w and from y=0 to y=h. 12 KTO Karatay University Engineering Faculty PHYS-‐102 Recitation Problem Set-‐2 13- A solid insulating sphere of radius a carries a net positive charge 3Q , uniformly distributed throughout its volume. Concentric with this sphere is a conducting spherical shell with inner radius b and outer radius c, and having a net charge ︎Q, as shown in Figure. (a) Construct a spherical gaussian surface of radius r>c and find the net charge enclosed by this surface. (b) What is the direction of the electric field at r>c? (c) Find the electric field at r>c. (d) Find the electric field in the region with radius r where c>r>b. (e) Construct a spherical gaussian surface of radius r, where c>r>b, and find the net charge enclosed by this surface. (f) Construct a spherical gaussian surface of radius r, where b>r>a, and find the net charge enclosed by this surface. (g) Find the electric field in the region b>r>a. (h) Construct a spherical gaussian surface of radius r<a, and find an expression for the net charge enclosed by this surface, as a function of r. Note that the charge inside this surface is less than 3Q. (i) Find the electric field in the region r<a. (j) Determine the charge on the inner surface of the conducting shell. (k) Determine the charge on the outer surface of the conducting shell. (l) Make a plot of the magnitude of the electric field versus r. 13 KTO Karatay University Engineering Faculty PHYS-‐102 Recitation Problem Set-‐2 14- An infinitely long cylindrical insulating shell of inner radius a and outer radius b has a uniform volume charge density ρ. A line of uniform linear charge density λ is placed along the axis of the shell. Determine the electric field everywhere. 14 KTO Karatay University Engineering Faculty PHYS-‐102 Recitation Problem Set-‐2 15- Two infinite, nonconducting sheets of charge are parallel to each other, as shown in Figure. The sheet on the left has a uniform surface charge density σ, and the one on the right has a uniform charge density -σ. Calculate the electric field at points (a) to the left of, (b) in between, and (c) to the right of the two sheets. 15 KTO Karatay University Engineering Faculty PHYS-‐102 Recitation Problem Set-‐2 16- A sphere of radius 2a is made of a nonconducting material that has a uniform volume charge density ρ. (Assume that the material does not affect the electric field.) A spherical cavity of radius a is now removed from the sphere, as shown in Figure. Determine the electric field within the cavity. 16 KTO Karatay University Engineering Faculty PHYS-‐102 Recitation Problem Set-‐2 17- A closed surface with dimensions a=b=0.400 m and c=0.600 m is located as in Figure. The left edge of the closed surface is located at position x=a. The electric field throughout the region is nonuniform and given by E=(3+2x2)î N/C where x is in meters. Calculate the net electric flux leaving the closed surface. What net charge is enclosed by the surface? 17 KTO Karatay University Engineering Faculty PHYS-‐102 Recitation Problem Set-‐2 18- An infinitely long insulating cylinder of radius R has a volume charge density that varies with the radius as ρ=ρ0 (a-r/b), where ︎ρ0, a, and b are positive constants and r is the distance from the axis of the cylinder. Use Gauss’s law to determine the magnitude of the electric field at radial distances (a) r<R and (b) r>R. 18 KTO Karatay University Engineering Faculty PHYS-‐102 Recitation Problem Set-‐2 19- A spherically symmetric charge distribution has a charge density given by ρ=a/r, where a is constant. Find the electric field as a function of r. 19 KTO Karatay University Engineering Faculty PHYS-‐102 Recitation Problem Set-‐2 20- A solid insulating sphere of radius R has a nonuniform charge density that varies with r according to the expression ρ=Ar2, where A is a constant and r<R is measured from the center of the sphere. Find the magnitude of the electric field (a) outside (r>R) (b) inside of the sphere. 20 KTO Karatay University Engineering Faculty PHYS-‐102 Recitation Problem Set-‐2 21- A solid, insulating sphere of radius a has a uniform charge density ρ and a total charge Q. Concentric with this sphere is an uncharged, conducting hollow sphere whose inner and outer radii are b and c, as shown in Figure. (a) Find the magnitude of the electric field in the regions r<a, a<r<b, b<r<c, and r>c. (b) Determine the induced charge per unit area on the inner and outer surfaces of the hollow sphere. 21 KTO Karatay University Engineering Faculty PHYS-‐102 Recitation Problem Set-‐2 22- For the configuration shown in Figure, suppose that a=5.00cm, b=20.0cm, and c=25.0cm. Furthermore, suppose that the electric field at a point 10.0 cm from the center is measured to be 3.60x103 N/C radially inward while the electric field at a point 50.0 cm from the center is 2.00x102 N/C radially outward. From this information, find (a) the charge on the insulating sphere, (b) the net charge on the hollow conducting sphere, and (c) the charges on the inner and outer surfaces of the hollow conducting sphere. 22

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