Question

In a particular system of units, a physical quantity can be expressed in terms of the electric charge e, electron mass m_e, Planck’s constant h, and Coulomb’s constant \(k=\frac{1}{4\pi\epsilon_0}\), where ε₀ is the permittivity of vacuum. In terms of these physical constants, the dimension of the magnetic field is [B] = [e]^α [m_e]^β [h]^γ [k]^δ. The value of α + β + γ + δ is ______.
 

Answer 1

Correct Answer: 4

\( [B] = [e]^\alpha [m_e]^\beta [h]^\gamma [K]^\delta \)

\( [B] = MT^{-2} \)

\( [e] = I T \)

\( [h] = ML^2 T^{-1} \)

\( [K] = ML^3 T^{-4} I^{-2} \)

Now, equating the dimensions:

\( MT^{-2} = [IT]^\alpha [M]^\beta \)

\( [ML^2 T^{-1}]^\gamma [ML^3 T^{-4} I^{-2}]^\delta \)

This results in the following system of equations:

\( 1 = \beta + \gamma + \delta \) (1)

\( -2 = \alpha - \gamma - 4\delta \) (2)

\( -1 = \alpha - 2\delta \) (3)

\( 0 = 2\gamma + 3\delta \) (4)

Solving equations (1), (2), (3), and (4), we get:

\( \alpha = 3 \)

\( \gamma = -3 \)

\( \delta = 2 \)

\( \beta = 2 \)

Finally, \( \alpha + \beta + \gamma + \delta = 4 \)