The dimensions of universal gravitational constant are:
[2004]
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 \([ML^{-2}T^{-2}A^2]\) belong to the:
[2022]
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.
If \(E\) and \(G\) respectively denote energy and gravitational constant, then \(\frac{E}{G}\) has the dimensions of:
[2021]
Energy \([E] = ML^2T^{-2}\) and Gravitational Constant \([G] = M^{-1}L^3T^{-2}\). So \([E/G] = M^2L^{-1}T^0\).
Dimensions of stress are:
[2020]
Stress is Force per unit Area. \([\text{Stress}] = [F]/[A] = (MLT^{-2})/L^2 = ML^{-1}T^{-2}\).
The dimensions of \((\mu_0\epsilon_0)^{-1/2}\) are:
[2012 Mains]
The speed of light \(c = 1/\sqrt{\mu_0\epsilon_0}\). Therefore, \((\mu_0\epsilon_0)^{-1/2} = c\). The dimensions of speed are \([LT^{-1}]\).
If \(C\) and \(R\) denote capacitance and resistance, the dimensional formula of \(CR\) is:
[1988]
The product \(CR\) is the time constant of an RC circuit. Its dimension is time, \(T\), which can be written as \(M^0L^0T^1\).
The dimensional formula of angular momentum is:
[1988]
Angular momentum is defined as \(L = r \times p\), where \(r\) is distance and \(p\) is linear momentum. Its dimensions are \([L][MLT^{-1}] = [ML^2T^{-1}]\).
If force \([F]\), acceleration \([A]\) and time \([T]\) are chosen as the fundamental physical quantities. Find the dimensions of energy.
[2021]
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, \(E = (F/A)(AT^2)^2T^{-2} = (F/A) A^2 T^4 T^{-2} = F A T^2\).
A physical quantity of the dimensions of length can be formed out of \(c\), \(G\) and \(e^2 / (4\pi\epsilon_0)\), where \(c\) is velocity of light, \(G\) is universal constant of gravitation and \(e\) is charge.
[2017, Delhi]
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: \(\frac{1}{c^2} \left( G \frac{e^2}{4\pi\epsilon_0} \right)^{1/2} = [L^{-2}T^2] \cdot ([M^{-1}L^3T^{-2}] \cdot [ML^3T^{-2}])^{1/2} = [L^{-2}T^2] \cdot ([L^6T^{-4}])^{1/2} = [L^{-2}T^2] \cdot [L^3T^{-2}] = [L]\).
The dimension of Planck constant equals to that of:
[2001]
Planck's constant \(h\) has dimensions \(ML^2T^{-1}\). Angular momentum also has dimensions \(ML^2T^{-1}\).