Electric Charges and Fields Set-3

Q1. Two point charges are placed on the $x$–axis: $+3.0\ \mu\mathrm{C}$ at $x=0$ and $+12.0\ \mu\mathrm{C}$ at $x=0.40\ \mathrm{m}$. At what point on the line between them (measured from $x=0$) is the net electric field zero?

Q2. The measured variation of the magnitude of electric field $E$ with distance $r$ from the centre of an object is: for $0\le r<R,\;E\propto r$ and for $r>R,\;E\propto 1/r^{2}$. Which of the following charge distributions best explains this $E(r)$ behaviour?

Q3. Two identical point charges $+2.0\ \mu\mathrm{C}$ are fixed at coordinates $(+a,0)$ and $(-a,0)$ with $a=0.04\ \mathrm{m}$. What is the magnitude of the resultant electric field at the point $(0,a)$? (Use $k=9.0\times10^{9}\ \mathrm{N\,m^{2}/C^{2}}$.)

Q4. Assertion (A): An isolated hollow conducting spherical shell is electrically neutral. If a point charge $+q$ is now placed at the geometric centre of the cavity, an induced charge $-q$ appears uniformly distributed over the inner surface and $+q$ appears on the outer surface.
Reason (R): The total induced charge on the inner surface equals $-q$ because the conductor was initially neutral and induced charges must sum to cancel the enclosed charge.

Q5. A thin circular ring of radius $R=0.10\ \mathrm{m}$ carries total charge $Q=5.0\ \mu\mathrm{C}$ uniformly. Find the magnitude of electric field on the axis of the ring at a point which is at a distance $x=R$ from the centre. (Use $k=9.0\times10^{9}\ \mathrm{N\,m^{2}/C^{2}}$.)

Q6. For a uniformly charged thin ring of radius $R$ and total charge $Q$, the axial electric field magnitude is $E(x)=\dfrac{kQx}{(R^{2}+x^{2})^{3/2}}$ for a point on the axis at distance $x$ from the centre. At which $x$ (measured from the centre) is $E(x)$ maximum?

Q7. A uniformly charged solid sphere of radius $R=0.40\ \mathrm{m}$ has unknown total charge $Q$. From an experimental $E(r)$ graph inside the sphere one reads $E(0.10\ \mathrm{m})=2.0\ \mathrm{N/C}$. Using $E_{\text{inside}}(r)=\dfrac{1}{4\pi\epsilon_0}\dfrac{Q\,r}{R^{3}}$, estimate $Q$ (take $1/4\pi\epsilon_0=9.0\times10^{9}\ \mathrm{N\,m^{2}/C^{2}}$).

Q8. Two very large parallel insulating sheets carry uniform surface charge densities $+\sigma$ and $-\sigma$ and face each other. If $\sigma=5.0\times10^{-6}\ \mathrm{C/m^{2}}$, what is the magnitude of the electric field in the region between the sheets? (Take $\epsilon_0=8.85\times10^{-12}\ \mathrm{C^{2}/(N\,m^{2})}$.)

Q9. Assertion (A): For a uniformly charged solid sphere of radius $R$ the electric field at an interior point a distance $r$ from the centre varies as $E\propto r$ (for $r<R$).
Reason (R): This linear variation arises because all the charge on a uniformly charged solid sphere is concentrated on its surface.

Q10. Two point charges $+Q$ and $-Q$ are fixed at $( -a,0,0)$ and $(+a,0,0)$ respectively. Consider a spherical Gaussian surface of radius $r$ centred at the origin with $r<a$ so that neither charge lies inside the surface. Which statement is correct?