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A wire loop ABCDE carrying a current I is placed in the x-y plane as shown in the figure. A particle of mass m and charge q is projected from origin with velocity vec(V) = (V_(0))/(sqrt(2))(hat(i) + hat(j)) m//sec. (a) Find the instaneous acceleration (b) If an external magnetic field vec(B) = B_(0) hat(i) is applied, find the force and torque acting on the loop due to this field. |
Answer» Solution :(a)  Magnetic field at `O` due to loop `ABCDE` `vec(B)_(1) = (mu_(0)I)/(4pi(r )/(2))(cos 45^(@) + cos 45^(@))(- hat(k))` `= -(mu_(0)I)/(2pi r).2.(1)/(sqrt(2))hat(k)` `= -(mu_(0)I)/(sqrt(2)pi r) hat(k)` `vec(B)_(2) = (mu_(0)I)/(2 r).(90^(@))/(360^(@))hat(k) = (mu_(0)I)/(8 r)hat(k)` `B_(0) = vec(B)_(1) + vec(B)_(2) = (mu_(0)I)/(r )((1)/(8) - (1)/(sqrt(2)pi)) hat(k)` Magnetic force on charged particle at `O` `vec(F) = q vec(V) xx vec(B)` `= q[(V_(0))/(sqrt(2))(hat(i) + hat(j))] xx [(mu_(0)I)/(r )((1)/(8) - (1)/(sqrt(2)pi))hat(k)]` `= (MU q V_(0) I)/(sqrt(2) r)((1)/(8) - (1)/(sqrt(2)pi))[(hat(i) + hat(j)) xx hat(k)]` `= (mu q V_(0) I)/(sqrt(2) r)((1)/(8) - (1)/(sqrt(2)pi))(-hat(j) + hat(i))` Acceleration `vec(a) = (vec(F))/(m)` `= (mu q V_(0) I)/(sqrt(2)m)((pi - 4sqrt(2))/(8 pi)) (hat(i) - hat(j))` `= (mu q V_(0)I)/(8sqrt(2) pi mr)(pi - 4sqrt(2))(hat(i) - hat(j))` (b) Now external magnetic field `vec(B) = B_(0) hat(i)` is applied. Net magnetic force on loop `= 0` (when a closed loop CARRYING single current is placed in uniform magnetic field, net magnetic force on loop is ZERO, whatever the shape of loop) AREA of loop `= (pi r^(2))/(4) - (1)/(2)r.(1)/(2) = r^(2)((pi)/(4) - (1)/(4))` Magnetic moment of loop `vec(M) = IA = (I r^(2))/(4)(pi - 1) hat(k)` Torque acting on loop `vec(tau) = vec(M) xx vec(B) = (B_(0)Ir^(2))/(4)(pi - 1)[hat(k) xx hat(i)]` `= (B_(0) I r^(2)(pi - 1))/(4) hat(j)` |
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