In Ampere's law `(oint (vec B).(vec dl))=(mu_0)I`, the current outside the curve is not included on the right hand side. Does it mean that the magnetic field B calculated by using Ampere's law , gives the contribution of only the currents crossing the area bounded by the curve?

Explain : "For some uses Ampere's circuital law ointvecB*vec(dl)=mu_(0)I is easy".

Consider a circular current-carrying loop of radius R in the x-y plane with centre at origin. Consider the line integral (L)=|int_(-L)^(L)vecB*vec(dl)| taken along z-axis. (a) Show that (L) monotonically increases with L. (b) Use an appropriate Amperian loop to show that (oo)=mu_(0)I , where I is the current in the wire. ( c) Verify directly the above result. (d) Suppose we replace the circular coil by a square coil of sides R carrying the same current I. What can you say about (L) and (oo) ?

Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass : (a) Show p=p_(i).+m_(i)V where p_(i) is the momentum of the ith particle (of mass m_(i) ) and p_(i).=m_(i)v_(i). . Note v_(i). is the velocity of the i^(th) particle relative to the centre of mass Also, prove using the definition of the centre of mass Sigmap_(i).=0 (b) Show K=K.+(1)/(2)MV^(2) where K is the total kinetic energy of the system of particles, K. is the total kinetic energy of the system when the particle velocities are taken with respect to the centre of mass and (1)/(2)MV^(2) is the kinetic energy of the translation of the system as a whole (i.e. of the centre of mass motion of the system). The result has been used in Sec. 7.14). (c ) Show vecL=vecL.+vecRxxvec(MV) where vecL.=Sigmavec(r_(i)).xxvec(p_(i)). is the angular momentum of the system about the centre of mass with velocities taken relative to the centre of mass. Remember vec(r._(i))=vec(r_(i))-vecR , rest of the notation is the velcities taken relative to the centre of mass. Remember vec(r._(i))=vec(r_(i))-vecR rest of the notation is the standard notation used in the chapter. Note vecL , and vec(MR)xxvecV can be said to be angular momenta, respectively, about and of the centre of mass of the system of particles. (d) Show vec(dL.)/(dt)=sumvec(r_(i).)xxvec(dp.)/(dt) Further, show that vec(dL.)/(dt)=tau._(ext) where tau._(ext) is the sum of all external torques acting on the system about the centre of mass. (Hint : Use the definition of centre of mass and Newton.s Thrid Law. Assume the internal forces between any two particles act along the line joining the particles.)

Verify the Ampere's law for magnetic field of a point dipole of dipole moment overset(to)(M) = M hat(k) . Take C as the closed curve running clockwise along (i) the z-axis from z=a gt 0 to z= R, (ii) along the quarter circle of radius R and centre at the origin, in the first quadrant of x-z plane, (iii) along the x-axis from x = R to x = a and (iv) along the quarter circle of radius a and centre at the origin in the first quadrant of x-z plane.

Do magnetic forces obey Newton's third law. Verify for two current elements vec(dl_(1))=dl(hati) located at the origin and vec(dl_(2))=dl(hatj) located at (0, R, 0). Both carry current I.