Two charges of same magnitude move in two circles of radii R, - R and R, -2R in a region
of constant uniform magnetic field B.
The work w, and W. done by the magnetic field in the two cases, respectively are such that:
(A) W, W, =0 (B) W, > W, (C) W = W, #0 (D) W, <W,
Answers
Answer:
Recall that in a static, unchanging electric field E the force on a particle with charge q will be:
F
=
qE
Where F is the force vector, q is the charge, and E is the electric field vector. Note that the direction of F is identical to E in the case of a positivist charge q, and in the opposite direction in the case of a negatively charged particle. This electric field may be established by a larger charge, Q, acting on the smaller charge q over a distance r so that:
E
=
∣
∣
F
q
∣
∣
=
k
∣
∣
qr
2
∣
∣
=
k
|
Q
|
r
2
It should be emphasized that the electric force F acts parallel to the electric field E. The curl of the electric force is zero, i.e.:
▽
×
E
=
0
A consequence of this is that the electric field may do work and a charge in a pure electric field will follow the tangent of an electric field line.
In contrast, recall that the magnetic force on a charged particle is orthogonal to the magnetic field such that:
F
=
qv
×
B
=
qvBsin
θ
where B is the magnetic field vector, v is the velocity of the particle and θ is the angle between the magnetic field and the particle velocity. The direction of F can be easily determined by the use of the right hand rule.
image
Right Hand Rule: Magnetic fields exert forces on moving charges. This force is one of the most basic known. The direction of the magnetic force on a moving charge is perpendicular to the plane formed by v and B and follows right hand rule–1 (RHR-1) as shown. The magnitude of the force is proportional to q, v, B, and the sine of the angle between v and B.
If the particle velocity happens to be aligned parallel to the magnetic field, or is zero, the magnetic force will be zero. This differs from the case of an electric field, where the particle velocity has no bearing, on any given instant, on the magnitude or direction of the electric force.
The angle dependence of the magnetic field also causes charged particles to move perpendicular to the magnetic field lines in a circular or helical fashion, while a particle in an electric field will move in a straight line along an electric field line.
A further difference between magnetic and electric forces is that magnetic fields do not net work, since the particle motion is circular and therefore ends up in the same place. We express this mathematically as:
W
=
∮
B
⋅
dr
=
0
Explanation:
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