For the two cells having same EMF E and internal resistance r, the current passing through the external resistor 6 \(\Omega\) is same when both the cells are connected either in parallel or in series. The value of internal resistance r is ___________ \(\Omega\).

In the given figure, an inductor and a resistor are connected in series with a battery of emf \(E\) volt. \(\frac{E^a}{2 b }\,J / s\) represents the maximum rate at which the energy is stored in the magnetic field (inductor). The numerical value of \(\frac{ b }{ a }\) will be ............

The length of a given cylindrical wire is increased to double of its original length. The percentage increase in the resistance of the wire will be..... \(\%\).

Two concentric circular coils, \(C _{1}\) and \(C _{2}\), are placed in the \(XY\) plane. \(C _{1}\) has \(500\) turns, and a radius of \(1\; cm . C _{2}\) has \(200\) turns and radius of \(20\, cm . C _{2}\) carries a time dependent current \(I(t)=\left(5 t^{2}-2 t+3\right) \;A\) where \(t\) is in \(s\). The \(emf\) nduced in \(C _{1}\) (in \(mV\) ), at the instant \(t =1\; s\) is \(\frac{4}{ x }\). The value of \(x\) is

A rigid wire consists of a semicircular portion of radius \(R\) and two straight sections. The wire is partially immerged in a perpendicular magnetic field \(B=B_0\ {j}\) as shown in figure. The magnetic force on the wire if it has a current \(i\) is _______.

A plane electromagnetic wave of frequency \(25\; \mathrm{GHz}\) is propagating in vacuum along the \(z-\)direction. At a particular point in space and time, the magnetic field is given by \(\overrightarrow{\mathrm{B}}=5 \times 10^{-8} \hat{\mathrm{j}}\; \mathrm{T}\). The corresponding electric field \(\overrightarrow{\mathrm{E}}\) is (speed of light  \(\mathrm{c}=3 \times 10^{8}\; \mathrm{ms}^{-1})\)

Statement \(I :\)Two forces \((\overrightarrow{{P}}+\overrightarrow{{Q}})\) and \((\overrightarrow{{P}}-\overrightarrow{{Q}})\) where \(\overrightarrow{{P}} \perp \overrightarrow{{Q}}\), when act at an angle \(\theta_{1}\) to each other, the magnitude of their resultant is \(\sqrt{3\left({P}^{2}+{Q}^{2}\right)}\), when they act at an angle \(\theta_{2}\), the magnitude of their resultant becomes \(\sqrt{2\left({P}^{2}+{Q}^{2}\right)}\). This is possible only when \(\theta_{1}<\theta_{2}\). Statement \(II :\) In the situation given above. \(\theta_{1}=60^{\circ} \text { and } \theta_{2}=90^{\circ}\) In the light of the above statements, choose the most appropriate answer from the options given below

A certain charge \(Q\) is divided into two parts \(q\) and \((Q-q) .\) How should the charges \(Q\) and \(q\) be divided so that \(q\) and \((Q-q)\) placed at a certain distance apart experience maximum electrostatic repulsion?

A block of mass \(0.1\, kg\) is connected to an elastic spring of spring constant \(640\, Nm^{-1}\) and oscillates in a damping medium of damping constant \(10^{-2}\, kg\,s^{-1}\) . The system dissipates its energy gradually. The time taken for its mechanical energy of vibration to drop to half of its initial value, is closest to ..... \(s\)

Two resistors are connected in series across a battery as shown in figure. If a voltmeter of resistance \(2000 \,\Omega\) is used to measure the potential difference across \(500 \,\Omega\) resister, the reading of the voltmeter will be ............... \(V\)

A Copper \((Cu)\) rod of length \(25\, {cm}\) and cross- sectional area \(3\, {mm}^{2}\) is joined with a similar Aluminium \((Al)\) rod as shown in figure. Find the resistance of the combination between the ends \(A\) and \(B\) (in \({m} \Omega\)) (Take Resistivity of Copper \(=1.7 \times 10^{-8}\, \Omega \,{m}\), Resistivity of Aluminium \(=2.6 \times 10^{-8}\, \Omega \,{m}\) )