NEET Dimensions MCQs & PYQs

From \(F = G \frac{m_1 m_2}{r^2}\), we get \(G = \frac{Fr^2}{m_1 m_2}\). So, \([G] = \frac{(MLT^{-2})(L^2)}{M^2} = M^{-1}L^3T^{-2}\).

The dimension of magnetic permeability is \(MLT^{-2}A^{-2}\). The given dimension \(ML^{-2}T^{-2}A^2\) does not match a standard physical quantity listed. Assuming a typo and it refers to magnetic permeability.

Energy \([E] = ML^2T^{-2}\) and Gravitational Constant \([G] = M^{-1}L^3T^{-2}\). So \([E/G] = M^2L^{-1}T^0\).

Stress is Force per unit Area. \([\text{Stress}] = [F]/[A] = (MLT^{-2})/L^2 = ML^{-1}T^{-2}\).

The speed of light \(c = 1/\sqrt{\mu_0\epsilon_0}\). Therefore, \((\mu_0\epsilon_0)^{-1/2} = c\). The dimensions of speed are \([LT^{-1}]\).

The product \(CR\) is the time constant of an RC circuit. Its dimension is time, \(T\), which can be written as \(M^0L^0T^1\).

Angular momentum is defined as \(L = r \times p\), where \(r\) is distance and \(p\) is linear momentum. Its dimensions are \([L][MLT^{-1}] = [ML^2T^{-1}]\).

We know \(E = ML^2T^{-2}\), \(F = MLT^{-2}\), \(A = LT^{-2}\), \(T = T\). From \(F = MA\), \(M = F/A\). Substitute \(M\) into \(E\): \(E = (F/A)L^2T^{-2}\). Also, \(L = AT^2\). So, \(E = (F/A)(AT^2)^2T^{-2} = (F/A) A^2 T^4 T^{-2} = F A T^2\).

Dimensions: \(c = [LT^{-1}]\), \(G = [M^{-1}L^3T^{-2}]\), and \(e^2/(4\pi\epsilon_0) = [ML^3T^{-2}]\). Let's check option d: \(\frac{1}{c^2} \left( G \frac{e^2}{4\pi\epsilon_0} \right)^{1/2} = [L^{-2}T^2] \cdot ([M^{-1}L^3T^{-2}] \cdot [ML^3T^{-2}])^{1/2} = [L^{-2}T^2] \cdot ([L^6T^{-4}])^{1/2} = [L^{-2}T^2] \cdot [L^3T^{-2}] = [L]\).

Planck's constant \(h\) has dimensions \(ML^2T^{-1}\). Angular momentum also has dimensions \(ML^2T^{-1}\).