Question

A length –scale (l) depends on the permittivity (ε) of a dielectric material ,Boltzmann constant (kB), the absolute temperature (T), the number per unit volume (n) of certain charged particles, and the charge (q) carried by each of the particles. Which of the following expression(s) for l is (are) dimensionally correct?

a. l = √(nq²/εkBT)
b. l = √(εkBT/nq²)
c. l = √(q²εn²/³kBT)
d. l = √(q²εn¹/³kBT)

Answer 1

Option b is the only dimensionally correct expression for the length scale (l) since it yields a dimension of length (L), making it consistent with the base quantities and their dimensions.

Step-by-step explanation:

To determine which expression(s) for the length scale (l) is dimensionally correct, it is essential to remember that the dimension of any physical quantity can be expressed as a product of powers representing the base quantities. The provided variables have the following dimensions in the International System of Units (SI):

• Permittivity (ε): M-1L-3T4I2
• Boltzmann constant (kB): M1L2T-2Θ-1
• Absolute temperature (T): Θ
• Number per unit volume (n): L-3
• Charge (q): I1T1

The correct expression for l must yield dimensions of length (L1). Let's assess the given options:

• a. l = √(nq2/εkBT) = √(L-3(IT)2/(M-1L-3T4I2ML2T-2Θ-1)) = √(M0L1T0I0Θ0) = L1/2
• b. l = √(εkBT/nq2) = √(M-1L-3T4I2ML2T-2Θ-1/L-3(IT)2) = √(M0L1T0I0Θ0) = L1/2
• c. l = √(q2εn2/3kBT) does not yield a dimension of length and is incorrect.
• d. l = √(q2εn1/3kBT) does not yield a dimension of length and is incorrect.

Therefore, option b. is the only dimensionally correct expression for the length scale l.