Nuclear & Particle Physics - Binding Energy

The binding energy of a heavy nucleus is about 7 million electron volts per nucleon, whereas the binding energy of a medium-weight nucleus is about 8 million electron volts per nucleon. Therefore, the total kinetic energy liberated when a heavy nucleus undergoes symmetric fission is most nearly

A. 1876 MeV
B. 938 MeV
C. 200 MeV
D. 8 MeV
E. 7 MeV

(GR9677 #64)

Solution:

KE = E_f  − E_i

Symmetric fission: the splitting of the nucleus into two fragments of approximately equal mass.

^AX  →  ^A₁Y  + ^A₂Z
with A₁ =  A₂ = A/2 and Y = Z  (for symmetric fission)

A E_i  → A₁ E_f 1 + A₂ E_f 2 = 2(A/2) E_f  = A E_f 

KE = A E_f  − A E_i  = A (8  − 7) MeV/nucleon = A MeV/nucleon

For heavy nucleus, A ≈ 200 nucleons. Example: ²³⁸U → A ≈ 238 nucleons

→ KE ≈ 200  MeV

Answer: C

Electromagnetism - Gauss' Law

If an electric field is given in a certain region by E_x = 0, E_y = 0, E_z = kz, where k is a nonzero constant, which of the following is true? 

A. There is a time-varying magnetic field.
B. There is charge density in the region.
C. The electric field cannot be constant in time.
D. The electric field is impossible under any circumstances.
E. None of the above.

(GR9277 #64)

Solution:

Gauss’ Law: ∇ ∙ E = ρ/ε₀

For E_x = 0, E_y = 0, E_z = kz

∇ ∙ E = ∂Ex/∂x + ∂Ey/∂y + ∂Ez/∂z = 0 + 0 + k = k

Since ∇ ∙ E ≠ 0 → There is charge density in the region.

Answer: B 

Electromagnetism - Impedance

An alternating current electrical generator has a fixed internal impedance R_g + jX_g and is used to supply power to a passive load that has an impedance R_g + jX₁, where j =√(−1). R_g ≠ 0 and X_g ≠ 0. For a maximum power transfer between the generator and the load, X₁ should be equal to

A. 0
B. X_g
C. −X_g
D. R_g
E. −R_g

(GR8677 #64)

Solution:

Maximum power transfer theorem (Impedance matching): 

Maximum power is transferred from a source to a load when

• Load resistance = the internal resistance of the source, or 
• Load impedance = complex conjugate source impedance (Z_l = Z_s^*)

Z_s = R_g + jX_g  with  j =√(−1)

Z_s^* = R_g − jX_g

For max power transferred: 
Z_l  = Z_s^*
X₁ = −X_g

Answer: C