If the work required to move the conductor shown in figure, one full turn in the positive direction at a rotational frequency `N` revolutions per minute, if `vec(B)=B_(0)hat(a)_(r)(B_(0)` is positive constant and `hat(a)_(r)` is a unit vector in radial direciton `) ,` is `y pi r B_(0)il`. Then `y` is .

Find the work and power required to move the conductor of length l shown in the fig. one full turn in the anticlockwise direction at a rotational frequency of n revolutions per second if the magnetic field is of magnitude B_(0) everywhere and points radially outwards from Z-axis. The figure shows that surface traced by the wire AB.

A magnetic field vec(B) = B_(0)hat(j) , exists in the region a ltxlt2a , and vec(B) = -B_(0) hat(j) , in the region 2a lt xlt 3a , where B_(0) is a positive constant . A positive point charge moving with a velocity vec(v) = v_(0) hat (i) , where v_(0) is a positive constant , enters the magnetic field at x= a . The trajectory of the charge in this region can be like

A magnetic field vec(B) = B_(0)hat(j) , exists in the region a ltxlt2a , and vec(B) = -B_(0) hat(j) , in the region 2a lt xlt 3a , where B_(0) is a positive constant . A positive point charge moving with a velocity vec(v) = v_(0) hat (i) , where v_(0) is a positive constant , enters the magnetic field at x= a . The trajectory of the charge in this region can be like

If vec(a)=4hat(i)-hat(j),vec(b)=-3hat(i)+2hat(j)and vec(c)=-hat(k). Then the unit vector hat( r) along the direction of sum of these vectors will be

If vec(a)=4hat(i)-hat(j),vec(b)=-3hat(i)+2hat(j)and vec(c)=-hat(k). Then the unit vector hat( r) along the direction of sum of these vectors will be

Motion in two dimensions, in a plane can be studied by expressing position, velocity and acceleration as vectors in cartesian coordinates A=A_(x)hat(i) + A_(y)hat(j) , where hat(i) and hat(j) are unit vector along x and y-directions, respectively and A_(x) and A_(y) are corresponding components of A. Motion can also be studied by expressing vectors in circular polar coordinates as A= A_(r)hat(r) + A_(theta) hat(theta) , where hat(r)=(r)/(r)=cos theta hat(i)+sin theta hat (j) and hat(theta)=-sin theta hat(i) + cos theta hat (j) are unit vectors along direction in which r and theta are increasing. (a) Express hat(i) and hat (j) in terms of hat(r) and hat (theta) . (b) Show that both hat(r) and hat(theta) are unit vectors and are perpendicular to each other. (c) Show that (d)/(dt)(hat(r))= omega hat(theta) , where omega=(d theta)/(dt) and (d)/(dt) (hat(theta))=-thetahat(r) . (d) For a particle moving along a spiral given by r=a theta hat(r) , where a = 1 (unit), find dimensions of a . (e) Find velocity and acceleration in polar vector representation for particle moving along spiral described in (d) above.

A particle is moving with a position vector, vec(r)=[a_(0) sin (2pi t) hat(i)+a_(0) cos (2pi t) hat(j)] . Then -

A particle is moving with a position vector, vec(r)=[a_(0) sin (2pi t) hat(i)+a_(0) cos (2pi t) hat(j)] . Then -