Moving Charges and Magnetism

Describe the motion of a charged particle in a cyclotron if the frequency of the radio frequency (rf) field were doubled.

The magnetic force depends on v which depends on the inertial frame of reference. Does then the magnetic force differ from inertial frame to frame? Is it reasonable that-the net acceleration has a different value indifferent frames of reference?

Show that a force that does no Work must be a velocity dependent force.

Five long wires A, B, C, D and E, each carrying current I are arranged to form edges of a pentagonal prism as shown in figure. Each carries current out of the plane of paper.
(a) What will be magnetic induction at a point on the axis 0? Axis is at a distance R from each wire.
(b) What will be the field if current in one of the wires (say A) is switched off?
(c) What if current in one of the wire (say A) is reversed?

A multirange current meter can be constructed by using a galvanometer circuit as shown in figure. We want a current meter that can measure 10 mA, 100 mA and 1 mA using a galvanometer of resistance 10 Ω  and that produces maximum deflection for current of 1 mA. Find S₁ ,S₂ and S₃ that have to be used.

Consider a circular current-carrying loop of radius R in the x-y plane with centre at origin. Consider the line integral ζ ( L ) = | ${\int }_{-\mathit{L}}^{\mathit{L}}\mathit{B}.\mathit{d}\mathit{l}$ | taken along z-axis.

(a) Show that ? ( ) monotonically increases with L.

(b) Use an appropriate Amperian loop to show that ζ ( ∞ ) = ?0? , 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 ζ ( L ) and ζ ( ∞ ) ?

This is a Long Answer Type Questions as classified in NCERT Exemplar

Explanation- B(z) points in the same direction on z-axis and hence , J(L) isa monotonically function of L so B.dl = B.dl as cos 0 = 1

(b) $\int B.dl={\mu }_{o}I$ but when L $\to$ $\infty$

So B $\propto$ 1/r³

(c) the magnetic field due to a circular current carrying loop of radius in the xy plane with centre at origin at any point lying at a distance of from origin.

B= $\frac{{\mu }_{0}I{R}^2}{2{(Z^2+R^2)}^{3/2}}$

Z=Rtan $\theta$

dz=Rsec² $\theta d\theta$

${\int }_{-\infty }^{\infty }{B}_{z}dz=\frac{{\mu }_{0}I}{2}{\int }_{-\pi /2}^{\pi /2}cos\theta d\theta$ = ${\mu }_{0}I$

Consider a circular current-carrying loop of radius R in the x-y plane with centre at origin. Consider the line integral ζ ( L ) = | ${\int }_{-\mathit{L}}^{\mathit{L}}\mathit{B}.\mathit{d}\mathit{l}$ | taken along z-axis.

(a) Show that �� ( ) monotonically increases with L.

(b) Use an appropriate Amperian loop to show that ζ ( ∞ ) = ��0�� , 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 ζ ( L ) and ζ ( ∞ ) ?

A uniform conducting wire of length 12a and resistance R is wound up as a current carrying coil in the shape of (i) an equilateral triangle of side a, (ii) a square of sides a, and (iii) a regular hexagon of sides a. The coil is connected to a voltage source V₀. Find the magnetic moment of the coils in each case.

An electron and a positron are released from (0, 0, 0) and (0, 0, 1.5R) respectively, in a uniform magnetic field B = B₀i, each with an equal momentum of magnitude p = eBR. Under what conditions on the direction of momentum will the orbits be non-intersecting circles?