JEE Mains · Physics · STD 12 - 1. Electric charges and fields
Charge is distributed within a sphere of radius \(R\) with a volume charge density \(\rho (r) = \frac{A}{{{r^2}}}{e^{ - 2r/a}}\) where \(A\) and \(a\) are constants. If \(Q\) is the total charge of this charge distribution, the radius \(R\) is.
- A \(a\,\log \,\left( {1 - \frac{Q}{{2\pi aA}}} \right)\)
- B \(\frac{a}{2}\,\log \,\left( {\frac{1}{{1 - \frac{Q}{{2\pi aA}}}}} \right)\)
- C \(a\,\log \,\left( {\frac{1}{{1 - \frac{Q}{{2\pi aA}}}}} \right)\)
- D \(\frac{a}{2}\,\log \left( {1 - \frac{1}{{2\pi aA}}} \right)\,\)
Answer & Solution
Correct Answer
(B) \(\frac{a}{2}\,\log \,\left( {\frac{1}{{1 - \frac{Q}{{2\pi aA}}}}} \right)\)
Step-by-step Solution
Detailed explanation
\(Q = \int \rho 4\pi {r^2}dr = \int_0^R {\left( {\frac{A}{{{r^2}}}{e^{ - \frac{{2r}}{a}}}} \right)} \left( {4\pi {r^2}} \right)dr\) \( = 4\pi A\frac{a}{2}\left( {1 - {e^{\frac{{ - 2R}}{a}}}} \right)\)…
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Physics
- Two cells are connected between points \(A\) and \(B\) as shown. Cell \(1\) has emf of \(12\,V\) and internal resistance of \(3\,\Omega\). Cell \(2\) has emf of \(6\,V\) and internal resistance of \(6\,\Omega\). An external resistor \(R\) of \(4\,\Omega\) is connected across \(A\) and \(B\). The current flowing through \(R\) will be \(.............A\).
JEE Mains 2023 Medium - At steady state the charge on the capacitor, as shown in the circuit below, is _________ \(\mu \mathrm{C}\).
JEE Mains 2025 Medium - A particle of mass \(m\) projected with a velocity ' \(u\) ' making an angle of \(30^{\circ}\) with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at its maximum height \(\mathrm{h}\) is _______.JEE Mains 2024 Hard
- Four particles, each of mass \(M\) and equidistant from each other, move along a circle of radius \(R\) under the action of their mutual gravitational attraction. The speed of each particle isJEE Mains 2014 Hard
- The angular speed of truck wheel is increased from \(900\, rpm\) to \(2460\, rpm\) in \(26\) seconds. The number of revolutions by the truck engine during this time is (Assuming the acceleration to be uniform).JEE Mains 2021 Medium
- Two charges \(\mathrm{q}_1\) and \(\mathrm{q}_2\) are separated by a distance of 30 cm. A third charge \(q_3\) initially at ' C ' as shown in the figure, is moved along the circular path of radius 40 cm from C to D. If the difference in potential energy due to movement of \(q_3\) from \(C\) to \(D\) is given by \(\frac{\mathrm{q}_3 \mathrm{~K}}{4 \pi \epsilon_0}\), the value of K is :
JEE Mains 2025 Easy
More PYQs from JEE Mains
- If the system of equations
\(\begin{aligned}
& 2 x-y+z=4 \\
& 5 x+\lambda y+3 z=12 \\
& 100 x-47 y+\mu z=212
\end{aligned}\)
has infinitely many solutions, then \(\mu-2 \lambda\) is equal toJEE Mains 2025 Easy - \(1\) mole of rigid diatomic gas performs a work of \(Q / 5\) when heat \(Q\) is supplied to it. The molar heat capacity of the gas during this transformation is \(\frac{ x R }{8},\) The value of \(x\) is \(\ldots \ldots \ldots .\) \([ K =\) universal gas constant \(]\)JEE Mains 2021 Hard
- An observer moves towards a stationary source of sound with a velocity equal to one-fifth of the velocity of sound. The percentage change in the frequency will be \(\dots \;\)%JEE Mains 2022 Medium
- The coefficient of \(x^{256}\) in the expansion of \((1-x)^{101}\left(x^{2}+x+1\right)^{100}\) is:JEE Mains 2021 Hard
- The distance of the point \((7,-3,-4)\) from the plane passing through the points \((2,-3,1),(-1,1,-2)\) and \((3,-4,2)\) is:JEE Mains 2023 Easy
- The earth's magnetic field lines resemble that of a dipole at the centre of the earth. If the magnetic moment of this dipole is close to \(8 \times 10^{22}\, Am^2\), the value of earth 's magnetic field near the equator is close to....\(Gauss\) (radius of the earth \(= 6.4 \times 10^6\, m\))JEE Mains 2013 Medium