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PHYS 196 Home Work 3 1. A thin wire of length 20cm carries a total charge of 24nC uniformly distributed over its length. Find the electric field at a point 6.0mm from the wire. (The distance of the point is so small the wire can be considered infinite in length.) 2. The diagram shows two parallel infinite line charges (charge density C / m ) a distance a apart. Find the magnitude of the electric field at the point P, which is on a plane perpendicular to the lines and is at distance a from one of the lines. a a 90º P 3. A 25cmx25cm square sheet carries a total charge of 40nC uniformly distributed over its area. Find the electric field at a point 3.0mm from the sheet. Find also the force on a 6.0nC charge placed at the point. (The point is so close to the sheet that the latter can be considered infinite.) 4. The diagram shows two parallel infinite sheet charges with surface charge densities and 3 respectively. Find the electric fields in the regions A, B and C. -σ 3σ A B C \ 5. A ring of radius a carrying a total charge q lies on the x-y plane with center at the origin. Find the direction and magnitude of the electric field at a point (0,0,z) on the z-axis. Find the value of z where this electric field is maximum. 6. A spherical shell of radius 25cm has a surface charge density 48nC / m 2 . Find the electric field at a point (a) just outside the surface, (b) in the interior, and (c) at a distance 50cm from the center. 7. A solid sphere of radius 25cm carries a total charge 72nC uniformly distributed over its volume. Find the electric field (a) at a point just outside the sphere, and (b) at an interior point 10cm from the center. 8. From a solid sphere of radius a with a uniform charge density , a spherical volume of radius a 2 is removed as shown. Find the magnitude and direction of the electric field at the center of the original sphere. 1 9. (Model of hydrogen molecule) The electron charge cloud in a hydrogen molecule is approximated by a solid sphere of radius R carrying uniform charge density so that the total charge is 2e .The two protons are taken to be point charges a distance 2a apart and collinear with the center of the sphere. Determine a so that the force on each proton is zero. 2a 10. A line charge of linear density extends to infinity from a perpendicular plane as shown. A point P on the plane is at a distance r from the point of intersection of the line and the plane. Use integration to find the components of the electric field at P parallel and perpendicular to the line. r P 11. A semicircular arc of radius a carries a uniform linear charge density . Use integration to find the electric field at the center. 12. Repeat the previous problem if the arc is 90 . 13. A hemispherical shell of radius a carries a uniform surface charge density . Use integration to find the electric field at the center. 14. A uniform electric field of magnitude 720N/C passes through a circular disk of radius 13cm . Find the electric flux if its face is (a) perpendicular to the field line, (b) at 30 to the field line, and (c) parallel to the field line. 15. A uniform electric field E is parallel to the axis of a hollow hemisphere of radius r . (a) What is the electric flux through the hemispherical surface? (b) What is the result if E is perpendicular to the axis? 16. A point charge q is at the center of a cube of side a . What is the electric flux through one of the faces? 17. Two concentric thin spherical shells of radius R1 and R2 with R1 R2 carry uniformly distributed charge Q1 and Q2 respectively. Use Gauss law to find the electric field at a point a distance r from the center, in the three cases r R2 , R2 r R1 and R1 r . 18. A solid sphere of radius a carries a uniform charge distribution over its volume. The total charge is Q . Use Gauss law to find the electric field at a point a distance r from the center in the two cases r a and ra. 2 19. The electron cloud in a hydrogen atom corresponds to a spherically symmetric charge distribution with volume charge density 0e 2 r a where r is the distance from the center. Find the electric field as a function of r . 20. Two coaxial infinitely long thin cylindrical shells of radius R1 and R2 with R1 R2 carry uniformly distributed charge per unit length of 1 and 2 respectively. Use Gauss law to find the electric field at a point a distance r from the axis, in the three cases r R2 , R2 r R1 and R1 r . Answers: 1. 3.6 105 N / C k 2. 10 a 3. 3.6 104 N / C 4. 0 , 2 0 , 0 , kqz a 5. , 3 2 2 a2 z 2 11. 2 k r 12. 13. k 14. 38.2 Nm2 / C, 19.1Nm2 / C , 0 2 15. r 2 E , 0 q 6 0 1 Q1 Q2 1 Q1 , , 0 17. 2 40 r 40 r 2 1 Q 1 Qr , 18. 2 40 r 40 a 3 6. 5400 N / C , 0, 1350 N / C 16. 7. 1.04 104 N / C, 4140 N / C 8. 2 k r a 6 0 9. R 2 19. 0 a 3 2r 2 2r 2 r a 1e 1 4 0 r 2 a 2 a 20. 1 1 2 , 20 r 10. k r 3 1 1 , 0 20 r