a) Let's assume a nuclear reactor was a completely isolated system, such that no matter particles can escape from it and the only thing that can get out is energy in the form of heat and radiation. If we weigh the whole reactor before switching it on, and again after it has been operating for a while, and find that the mass difference is 0.05 grams, what is the total amount of energy that the reactor has produced?

b)Nuclear reactions can produce electrically charged particles (of say mass m and charge q). To measure the kinetic energy E of such a particle we want to find its speed v, and one way to do this in practice is to use a magnetic field (of constant magnitude B, and perpendicular to the particle's direction of motion). The Lorentz force law tells us that such a particle traveling though the magnetic field will experience a force of magnitude q v B. What direction does the force point in? Explain why this causes the particle to move in a circle, and derive a formula that gives the kinetic energy of the particle in terms of the (observable) radius of this circle.

this sso guy is nuts if he thinks we should be able to do this...

a)

E = m c^2
here
m = .05 grams = .05*10^-3 kg = 5*10^-5 kg
c = 3*10^8m/s so c^2 = 9*10^16 m^2/s^2
so
E = 45 * 10^11 = 4.5*10^12 Joules

b) Force perpendicular to velocity gives centripetal acceleration
F = m v^2/R
so
q v B = m v^2/R
so
v = (q/m) B R
Ke = (1/2) m v^2
Ke = (1/2) m (q/m)^2 B^2 R^2
Ke = (1/2) (q B R)^2 sqrt(m)

a) To find the total amount of energy produced by the reactor, we can use Einstein's mass-energy equivalence principle, expressed by the famous equation E=mc². Here's how we can calculate it step by step:

1. Determine the mass difference: Weighing the reactor before switching it on and again after it has been operating for a while allows us to find the mass difference. In this case, the mass difference is given as 0.05 grams.

2. Convert the mass difference to kilograms: To use the equation E=mc², we need the mass in kilograms. Therefore, we convert the mass difference of 0.05 grams to 0.00005 kilograms.

3. Calculate energy using Einstein's equation: Now, we can plug the value of mass (m) into the equation E=mc². Since c (the speed of light) is a constant, we can multiply the mass by the square of the speed of light to calculate the energy. The speed of light is approximately 3 × 10^8 meters per second.

E = (0.00005 kg) × (3 × 10^8 m/s)²
= 0.00005 kg × 9 × 10^16 m²/s²
= 4.5 × 10^12 Joules

Therefore, the total amount of energy produced by the reactor is approximately 4.5 × 10^12 Joules.

b) The Lorentz force law describes the force experienced by a charged particle moving through a magnetic field. According to the law, the force (F) acting on the particle is given by the equation F = qvB, where q is the charge of the particle, v is its velocity, and B is the magnetic field strength.

To determine the direction of the force, we can use the right-hand rule for positive charges. If we point our thumb in the direction of the particle's velocity (v) and our fingers in the direction of the magnetic field (B), then the force (F) will be perpendicular to both, and it will be either upwards or downwards depending on the charge of the particle.

In this case, assuming a positively charged particle, the force will be perpendicular to both the velocity and the magnetic field, pointing either upwards or downwards.

This force causes the particle to move in a circle because it acts as a centripetal force, directed towards the center of the circular path. The centripetal force is given by the equation F = (mv²)/r, where m is the particle's mass and r is the radius of the circular path.

Setting the Lorentz force (qvB) equal to the centripetal force ((mv²)/r), we can derive a formula for the kinetic energy of the particle in terms of the radius:

qvB = (mv²)/r

Rearranging the equation, we can solve for the kinetic energy (E):

E = (1/2)mv²
= (1/2)qvrB

Now, substituting the value of the centripetal force (mv²)/r with qvB:

E = (1/2)(qvB)r
= (1/2)qvrB

Therefore, the kinetic energy (E) of the particle can be determined as half the product of the charge (q), velocity (v), magnetic field strength (B), and radius (r) of its circular path.