Question

Week 4 Physics 272 Recitation, Spring 2019 Problem: You and your friend are discussing last week's recitation problem, in which an electron falling down a tunnel through a uniformly charged earth exhibits simple harmonic motion. Instead of being uniformly charged, imagine that the Earth is given a spherically symmetric charge density p(r)-b/r, where b is some constant. If an electron is released from rest inside this tunnel at some initial distance from the center, find an expression for the acceleration of the electron inside the tunnel (ignore gravity). If the electron is released from rest at the surace o b will ensure that the electron reaches the center of the Earth in 12 seconds? USE THIS PAGE AND THE BACK FOR SCRATCH PAPER

Answer 1

If _$\rho$ is the density of Earth, then the mass of sphere will be given by -

m = _$\rho$ V

We know that, volume of a sphere is given by -

V = (4/3)$\pi$r³

Then, we have

m = (4/3)$\pi$r³_$\rho$

Using a universal gravitation formula, we have

F = G M m / r²

F = G M (4/3)$\pi$r³_$\rho$ / r²

F = (4$\pi$/3) G _$\rho$ r M

If an electron is released from rest inside this tunnel at some initial distance from the center, then an expression for the acceleration of an electron inside the tunnel which will be given as -

a = F / M $\Rightarrow$ [(4$\pi$/3) G _$\rho$ r M] / M

a = (4$\pi$/3) G _$\rho$ r

where, _$\rho$ = symmetric charge density = b / r

then, we get

a = (4$\pi$/3) G (b / r) r

a = (4$\pi$/3) G b

If the electron is released from rest at the surface, then the value of 'b' which will be given by -

T = 2$\pi$_$\sqrt{}$r / a

T = 2$\pi$_$\sqrt{}$r / [(4$\pi$/3) G b]

T = 2$\pi$_$\sqrt{}$3 r / (4$\pi$) G b

T = _$\sqrt{}$3 r / G b

squaring on both sides & we have

T² = 3 r / G b

b = 3 r / G T²

where, r = distance from center of Earth to the electron = 6.371 x 10⁶ m

G = gravitational constant = 6.67 x 10^-11 Nm²/kg²

T = time taken by the electron = 12 sec

then, we get

b = (3 x 6.371 x 10⁶ m) / [(6.67 x 10^-11 Nm²/kg²) (12 s)²]

b = [(19113000 m) / (9.6048 x 10^-9 m³/kg)]

b = 1.98 x 10¹⁵ kg/m²