The diffusion equation is given by:
\(\dfrac{\partial n}{\partial t}=D\dfrac{\partial^2 n}{\partial x^2}\)
where \(n(x,t)\) is the concentration of particles per unit volume at position \(x\), at time \(t\). The dimensions of the diffusion constant \(D,\) are:
1. \(LT^{-1}\)
2. \(L^{-1}T\)
3. \(L^{5}T^{-1}\)
4. \(L^{2}T^{-1}\)
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Which of the following equations is dimensionally correct?
\((I)~~ v=\sqrt{\dfrac{P}{\rho}}~~~~~~(II)~~v=\sqrt{\dfrac{mgl}{I}}~~~~~~(III)~~v=\dfrac{Pr^2}{2\eta l}\)
(where \(v=\) speed, \(P=\) pressure; \(r,\) \(l\) are lengths; \(\rho=\) density, \(m=\) mass, \(g=\) acceleration due to gravity, \(I=\) moment of inertia, and \(\eta=\) coefficient of viscosity)
| 1. | \(I~ \text{and}~II\) |
| 2. | \(I~ \text{and}~III\) |
| 3. | \(II~ \text{and}~III\) |
| 4. | \(I,~II~\text{and}~III\) |
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The acceleration due to gravity on the surface of the Earth is \(g=10~\text{m/s}^2\) . The value in \(\text {km/minute}^2\) is:
| 1. | \(36\) | 2. | \(0.6\) |
| 3. | \(\dfrac{10}{6}\) | 4. | \(3.6\) |
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Find the unit of the ratio \(\dfrac{E}{gB}\) where \(E\) = electric field, \(B\) = magnetic field, \(g\) = acceleration due to gravity.
1. \(\text{s}^{-1}\)
2. \(\text{s}\)
3. \(\text{m/s}\)
4. \(\text{s/m}^2\)
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\(S=\int_{0}^{t}(K-U)^{} d t\)
where \(K\) is the kinetic energy, \(U\) is the potential energy and the integral is over the time, \(t\) during the motion.
The proper unit of action will be:
| 1. | kg-m/s | 2. | kg-m2 |
| 3. | kg-m2-s | 4. | kg-m2/s |
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\(h\) : Planck's constant, \(K\) : kinetic energy, \(\omega\) : angular speed/frequency, \(F\): force, \(L\) : inductance, \(i\) : current, \(q\) : charge, \(t\) : time, \(x\): distance
1. \(h, ~Ftx,~ Liq\)
2. \(K , h\omega , \omega Li\)
3. \(Fx, ~Li^2,~K \omega\)
4. \(\dfrac{Fx}{t} ,~ Kx, ~ ht\)
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| (A) | \(\dfrac{\text{(Magnetic flux)}^2}{\text{Electrical resistance}}\) | (B) | \(\text{Torque}\times\text{time}\) |
| (C) | \(\text{Momentum}\times\text{length}\) | (D) | \(\dfrac{\text{Power}}{\text{time}}\) |
2. A, B, C
3. B, D
4. A, B, C, D
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1. \(\Omega\) (ohm)
2. s/m
3. \(\text{H }\)(henry)
4. \(\text{F }\) (farad)
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where \(\mu_0\) is the permeability of free space and \(\sigma\) is the electrical conductivity (of a plasma). \(\eta\) is referred to as the magnetic diffusivity. Its SI unit is:
1. m2/s
2. m/s2
3. C-m2/s
4. T-m/s
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Let the two quantities be defined as follows:
\(\int \vec{B}.\vec{dl} = W\)and \(\int \vec{E}.\vec{dS} = U\)
where \(\vec{ B}\) is the magnetic field, \(d\vec{l}\) is a small length, \(\vec{ E}\) is the electric field and \(d\vec{S}\) is a small area. The ratio \(\dfrac{U}{W}\) has the dimensions of:
1. \(\left[L^2T^{-2}\right]\)
2. \(\left[LT^{-2}\right]\)
3. \(\left[L^2T^{-1}\right]\)
4. \(\left[LT^{-1}\right]\)
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