Charges \(Q _{1}\) and \(Q _{2}\) arc at points \(A\) and \(B\) of a right angle triangle \(OAB\) (see figure). The resultant electric field at point \(O\) is perpendicular to the hypotenuse, then \(Q _{1} / Q _{2}\) is proportional to

Given below are two statements:
Statement I: An object moves from position \( r_{1} \) to position \( r_{2} \) under a conservative force field \( \vec{F} \). The work done by the force is \( W=-\int_{r_{1}}^{r_{2}}\vec{F}\cdot d\vec{r} \).
Statement II Any object moving from one location to another location can follow infinite number of paths. Therefore, the amount of work done by the object changes with the path it follows for a conservative force.
In the light of above statement, choose the correct answer from the options given below:

The source of time varying magnetic field may be \((A)\) a permanent magnet \((B)\) an electric field changing linearly with time \((C)\) direct current \((D)\) a decelerating charge particle \((E)\) an antenna fed with a digital signal Choose the correct answer from the options given below:

For three low density gases \(A, B, C\) pressure versus temperature graphs are plotted while keeping them at constant volume, as shown in the figure. The temperature corresponding to the point ' \(K\) ' is \(...........\,{}^{\circ}\,C\)

A body is in simple harmonic motion with time period half second \((T\, = 0.5\, s)\) and amplitude one \(cm\, (A\,= 1\, cm)\). Find the average velocity in the interval in which it moves form equilibrium position to half of its amplitude .... \(cm/s\)

Two charges \(7 \mu C\) and \(-2 \mu C\) are placed at \((-9,0,0) cm\) and \((9,0,0) cm\) respectively in an external field \(E=\frac{ A }{ r ^2} \hat{ r }\), where \(A=9 \times 10^5 N / C . m ^2\).
Considering the potential at infinity is 0 , the electrostatic energy of the configuration is ___________ J .

If three moles of monoatomic gas \(\left(\gamma=\frac{5}{3}\right)\) is mixed with two moles of a diatomic gas \(\left(\gamma=\frac{7}{5}\right)\), the value of adiabatic exponent \(\gamma\) for the mixture is _______.

A force \(\left(3 x^2+2 x-5\right) N\) displaces a body from \(\mathrm{x}=2 \mathrm{~m}\) to \(\mathrm{x}=4 \mathrm{~m}\). Work done by this force is _______ J.

In free space, a particle \(A\) of charge \(1\,\mu C\) is held fixed at a point \(P.\) Another particle \(B\) of the same charge and mass \(4\,\mu g\) is kept at a distance of \(1\,mm\) from \(P\). If \(B\) is released, then its velocity at a distance of \(9\,mm\) from \(P\) is [ Take \(\frac{1}{{4\pi {\varepsilon _0}}} = 9 \times {10^9}\,N{m^2}{C^{ - 2}}\) ]

In the given circuit the \({AC}\) source has \(\omega=100\, {rad} {s}^{-1}\). Considering the inductor and capacitor to be ideal, what will be the current \(I\) flowing through the circuit? (in \(A\))

A particle of charge \(q\) and mass \(m\) is projected from origin with an initial velocity \(\vec{v} = \left(\dfrac{v_0}{\sqrt{2}}\hat{x} + \dfrac{v_0}{\sqrt{2}}\hat{y}\right)\). There exists a uniform magnetic field \(\vec{B} = B_0 \hat{z}\) and a space varying electric field \(\vec{E} = E_0 e^{-\lambda x}\hat{x}\) within the region \(0 \leq x \leq L\). After travelling a distance such that \(x\)-coordinate has changed from \(x=0\) to \(x=L\), the change in the kinetic energy is _______.

The temperature of an open room of volume \(30\ m^3\) increases from \(17^o C\) to \(27vC\) due to sunshine. The atmospheric pressure in the room remains \(1 \times 10^5\ Pa\). Ifni and nr are the number of molecules in the room before and after heating, then \(n_f-n_i\) will be :

A uniform cylindrical rod of length \(L\) and radius \(r\), is made from a material whose Young's modulus of Elasticity equals \(Y\). When this rod is heated by temperature \(T\) and simultaneously subjected to a net longitudinal compressional force \(F\), its length remains unchanged. The coefficient of volume expansion, of the material of the rod, is (nearly) equals to