Question

A heavy block is attached to the ceiling by a spring that has a force constant ‘k’. A conducting rod is attached to block. The combined mass of the block and the rod is m. The rod can slide without friction along two vertical parallel rails, which are a distance L apart. A capacitor of known capacitance C is attached to the rails by the wires. The entire system is placed in a uniform magnetic field B. Find the time period T of the vertical oscillations of the block. Neglect the electrical resistance of the rod and all wires

• A. $2 \pi \sqrt { \frac { \mathrm {~m} + \mathrm { CB } ^ { 2 } \mathrm {~L} ^ { 2 } } { \mathrm { k } } }$
• B. $2 \pi \sqrt { \frac { \mathrm { m } ^ { 2 } + \mathrm { CBL } } { \mathrm { k } } }$
• C. $4 \pi \sqrt { \frac { \mathrm { m } ^ { 2 } + \mathrm { CB } ^ { 2 } \mathrm {~L} ^ { 2 } } { \mathrm { k } } }$
• D. None of these

Answer 1

Answer: (A)

$2 \pi \sqrt { \frac { \mathrm {~m} + \mathrm { CB } ^ { 2 } \mathrm {~L} ^ { 2 } } { \mathrm { k } } }$

Answer 2

Using Kirchoff ’s equation

Blv = 0 [Where ]

q = CBlv

i = $\frac { \mathrm { dq } } { \mathrm { dt } }$ = CBl $\frac { \mathrm { dv } } { \mathrm { dt } }$

Magnetic force on AB bar or block = Bil

F_mag = B²l²C $\frac { \mathrm { dv } } { \mathrm { dt } }$

For initial equilibrium,

kx = mg ... (1)

mg k(x + y) B²l²C= ma ... (2)

mg kx ky B²l²Ca = ma

a = $\frac { - \mathrm { k } } { \mathrm { m } + \mathrm { B } ^ { 2 } \ell ^ { 2 } \mathrm { Ca } } \mathrm { y }$

Comparing equation of a by a = w²y

w =

T = $2 \pi \sqrt { \frac { \mathrm {~m} + \mathrm { CB } ^ { 2 } \mathrm {~L} ^ { 2 } } { \mathrm { k } } }$