A uniform magnetic field of 1.5 T exists in a cylindrical region ofradius10.0 cm, its direction parallel to the axis along east to west. Awire carrying current of 7.0 A in the north to south direction passesthrough this region. What is the magnitude and direction of theforce on the wire if, (a) the wire intersects the axis, (b) the wire is turned from N-S to northeast-northwest direction, (c) the wire in the N-S direction is lowered from the axis by a distanceof 6.0 cm?
Given: The radius of cylindrical region is 10 cm , the magnitude of the magnetic field is 1.5 T and the magnitude of the current in the wire is 7 A which is flowing from north to south direction.
(a)
If the wire intersects the axis, then the length of the wire is diameter of the cylindrical region.
l = 2 r (1)
Where, the length of the wire is l and the radius of the wire is r .
By substituting the values in equation (1), we get
l = 2 × 0.1 = 0.2 m
The magnetic force acting on the wire is given as,
F = B I l sin θ (2)
Where, the magnetic force is F , the magnetic field is B , the current is I and the angle between the current and magnetic field is θ .
By substituting the given values in equation (2), we get
F = 1.5 × 7 × 0.2 × sin 90 ° = 10.5 × 0.2 × 1 = 2.1 N
Using Fleming’s left-hand rule, the force acts in vertically downward direction.
Thus, the magnitude of force is 2.1 N in vertically downward direction.
(b)
The length of the wire when the direction is changed to Northeast-Northwest is given as,
l 1 = l sin θ
Where, the new length of the wire is l 1 .
By substituting the given values in equation (2), we get
F = 1.5 × 7 × 0.2 sin 45 ° × sin 45 ° = 1.5 × 7 × 0.2 = 2.1 N
Thus, the magnitude of force is 2.1 N in vertically downward direction.
(c)
The new length of wire is given as,
( l 2 2 ) 2 = 4 ( d + r ) (3)
Where, the new length of the wire is l 2 , the distance below the ground is d and radius is r .
By substituting the given values in equation (3), we get
( l 2 2 ) 2 = 4 ( 10 + 6 ) ( l 2 2 ) 2 = 4 × 16 ( l 2 2 ) 2 = 64 ( l 2 2 ) = 8
Further simplify the above expression.
l 2 = 16 cm = 0 .16 m
By substituting the given values in equation (2), we get
F = 1.5 × 7 × 0.16 = 1.68 N
Thus, the magnitude of force is 1.68 N in vertically downward direction.
Given: The radius of cylindrical region is
(a)
If the wire intersects the axis, then the length of the wire is diameter of the cylindrical region.
Where, the length of the wire is
By substituting the values in equation (1), we get
The magnetic force acting on the wire is given as,
Where, the magnetic force is
By substituting the given values in equation (2), we get
Using Fleming’s left-hand rule, the force acts in vertically downward direction.
Thus, the magnitude of force is
(b)
The length of the wire when the direction is changed to Northeast-Northwest is given as,
Where, the new length of the wire is
By substituting the given values in equation (2), we get
Thus, the magnitude of force is
(c)
The new length of wire is given as,
Where, the new length of the wire is
By substituting the given values in equation (3), we get
Further simplify the above expression.
By substituting the given values in equation (2), we get
Thus, the magnitude of force is
