Question

[1] A rod of negligible thickness lies on the x-axis with one end at x=0 and another at x=L. Positive charges are distributed along the length of the rod according to the linear charge density ?(x) = ?? x/L, with ?? > 0 as a constant. Point P on the x-axis is a distance x? away from the origin and is our point of observation. Assume x? > L. (A) Sketch the situation described above, paying attention to approximate charge distribution at the two ends of the rod. Clearly show where point P is with respect to the rod. (B) Calculate the electric field ? at location P. This calculation involves multiple steps because the charges are not point charges but are distributed over a length! For point charges we can write down E=kq/r² but for distributed charges we have to first reduce the problem to point charge and then count up to include contribution of all charges. Hint: (i) First set up an expression for d?, the electric field due to an infinitesimally small amount of charge on the rod and show that dE = k??/L * x dx / (x?-x)². For inspiration (note: situation is different there) see Example 21-11 in the textbook. (ii) Next, argue that the total electric field at location P is given by Enet = k??/L ??? x dx / (x?-x)². The physics problem is now a calculus problem where you have to solve the integral! (iii) If given that the indefinite integral ? z dz / (a-z)² = a/(a-z) + ln(a-z) + constant, where a>0, calculate the electric field Enet at P. (iv) What is the direction of Enet? (C) Calculate in terms of ??, L and other constants, the total charge Q on the rod. Hint: by definition, the total charge=? ?(x) dx, with appropriate limits on the integral. [2] In the figure, an object of mass m and positive charge q is suspended in static equilibrium due to the combined effect of tension T in the string attached to the object and force F due to an inclined E field (inclined at 30° to the horizontal) in space. (A) Calculate the magnitude of ?, in terms of m, q, and numerical constants. (B) Find the magnitude of T in terms of the same quantities. Take g= 10 m/s². E field lines T Q F mg 30°

          [1] A rod of negligible thickness lies on the x-axis with one end at x=0 and another at x=L. Positive charges are distributed along the length of the rod according to the linear charge density ?(x) = ?? x/L, with ?? > 0 as a constant. Point P on the x-axis is a distance x? away from the origin and is our point of observation. Assume x? > L.

(A) Sketch the situation described above, paying attention to approximate charge distribution at the two ends of the rod. Clearly show where point P is with respect to the rod.

(B) Calculate the electric field ? at location P. This calculation involves multiple steps because the charges are not point charges but are distributed over a length! For point charges we can write down E=kq/r² but for distributed charges we have to first reduce the problem to point charge and then count up to include contribution of all charges. Hint:
(i) First set up an expression for d?, the electric field due to an infinitesimally small amount of charge on the rod and show that dE = k??/L * x dx / (x?-x)². For inspiration (note: situation is different there) see Example 21-11 in the textbook.
(ii) Next, argue that the total electric field at location P is given by Enet = k??/L ??? x dx / (x?-x)². The physics problem is now a calculus problem where you have to solve the integral!
(iii) If given that the indefinite integral ? z dz / (a-z)² = a/(a-z) + ln(a-z) + constant, where a>0, calculate the electric field Enet at P.
(iv) What is the direction of Enet?

(C) Calculate in terms of ??, L and other constants, the total charge Q on the rod. Hint: by definition, the total charge=? ?(x) dx, with appropriate limits on the integral.

[2] In the figure, an object of mass m and positive charge q is suspended in static equilibrium due to the combined effect of tension T in the string attached to the object and force F due to an inclined E field (inclined at 30° to the horizontal) in space. (A) Calculate the magnitude of ?, in terms of m, q, and numerical constants. (B) Find the magnitude of T in terms of the same quantities. Take g= 10 m/s². 

E field lines

T

Q

F

mg

30°

[1] A rod of negligible thickness lies on the x-axis with one end at x=0 and another at x=L. Positive charges are distributed along the length of the rod according to the linear charge density ?(x) = ?? x/L, with ?? > 0 as a constant. Point P on the x-axis is a distance x? away from the origin and is our point of observation. Assume x? > L.

(A) Sketch the situation described above, paying attention to approximate charge distribution at the two ends of the rod. Clearly show where point P is with respect to the rod.

(B) Calculate the electric field ? at location P. This calculation involves multiple steps because the charges are not point charges but are distributed over a length! For point charges we can write down E=kq/r² but for distributed charges we have to first reduce the problem to point charge and then count up to include contribution of all charges. Hint:
(i) First set up an expression for d?, the electric field due to an infinitesimally small amount of charge on the rod and show that dE = k??/L * x dx / (x?-x)². For inspiration (note: situation is different there) see Example 21-11 in the textbook.
(ii) Next, argue that the total electric field at location P is given by Enet = k??/L ??? x dx / (x?-x)². The physics problem is now a calculus problem where you have to solve the integral!
(iii) If given that the indefinite integral ? z dz / (a-z)² = a/(a-z) + ln(a-z) + constant, where a>0, calculate the electric field Enet at P.
(iv) What is the direction of Enet?

(C) Calculate in terms of ??, L and other constants, the total charge Q on the rod. Hint: by definition, the total charge=? ?(x) dx, with appropriate limits on the integral.

[2] In the figure, an object of mass m and positive charge q is suspended in static equilibrium due to the combined effect of tension T in the string attached to the object and force F due to an inclined E field (inclined at 30° to the horizontal) in space. (A) Calculate the magnitude of ?, in terms of m, q, and numerical constants. (B) Find the magnitude of T in terms of the same quantities. Take g= 10 m/s².

E field lines

T

Q

F

mg

30°

Added by Donna V.

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