Exams › JEE Advanced › Physics

A characteristic length l is built from the permittivity (epsilon) of a dielectric, the Boltzmann constant kB, the absolute temperature T, the number density n of charged particles (number per unit volume), and the charge q on each particle. Which of the following expressions for l is dimensionally correct?

1. l = sqrt(epsilon*kB*T/(n*q²))
2. l = sqrt(n*q²/(epsilon*kB*T))
3. l = sqrt(q²/(epsilon*n^(2/3)*kB*T))
4. l = sqrt(q²/(epsilon*n^(1/3)*kB*T))

Correct answer: l = sqrt(epsilon*kB*T/(n*q²))

Solution

From Coulomb energy, q²/(epsilon) has dimensions of (energy * length). So epsilon*kB*T/q² has dimensions of (energy)/(energy*length per...) -> work it out: epsilon*kB*T/q² has dimension 1/length, and dividing by n (1/length³) gives length². The square root is a length. This is the Debye screening length, l = sqrt(epsilon kB T/(n q²)).

Related JEE Advanced Physics questions

• The diameter of a cylinder is measured using a Vernier callipers with no zero error. It is found that the zero of the Vernier scale lies between 5.10 cm and 5.15 cm of the main scale. The Vernier scale has 50 divisions equivalent to 2.45 cm. The 24th division of the Vernier scale exactly coincides with one of the main scale divisions. The diameter of the cylinder is
• In the context of electromagnetic theory, the dimensions of electric and magnetic fields are interconnected. If [E] represents the dimension of the electric field and [B] represents the dimension of the magnetic field, which of the following correctly describes their relationship?
• What is the mathematical relationship between the dimensions of [ε0] and [μ0]?
• The ratio r = (1 - a) / (1 + a) is derived from the measurement of a dimensionless parameter a. If the uncertainty in a is represented as Δa (where Δa / a << 1), what is the uncertainty Δr in calculating r?
• In a specific unit system, a magnetic field's dimensions can be represented using the electric charge e, the mass of an electron mₑ, Planck's constant h, and Coulomb's constant k = 1 / (4πϵ₀), where ϵ₀ represents the vacuum permittivity. If the dimensions of the magnetic field [B] are expressed as [e]ᵅ [mₑ]ᵝ [h]ᵞ [k]ᵟ, what is the sum of the exponents α, β, γ, and δ?
• A screw gauge is used to measure the diameter of a wire. The main scale has a pitch of 0.5 mm, and the circular scale is divided into 100 equal parts. A full rotation of the circular scale moves the main scale by two divisions. The following readings were recorded during the measurements: Measurement condition | Main scale reading | Circular scale reading When gauge arms touch without the wire | 0 divisions | 4 divisions First measurement with wire | 4 divisions | 20 divisions Second measurement with wire | 4 divisions | 16 divisions Based on these readings, what are the diameter and cross-sectional area of the wire as determined by the screw gauge?

⚔️ Practice JEE Advanced Physics free + battle 1v1 →