Two large vertical and parallel metal plates having a separation of $$1$$ $$cm$$ are connected to a $$DC$$ voltage source of potential difference $$X$$. A proton is released at rest midway between the two plates. It is found to move at $${45^ \circ }$$ to the vertical JUST after release. Then $$X$$ is nearly

Consider a thin spherical shell of radius $$R$$ with center at the origin, carrying uniform positive surface charge density. The variation of the magnitude of the electric field $$\left| {\overrightarrow E \left( r \right)} \right|$$ and the electric potential $$V(r)$$ with the distance $$r$$ from the center, best represented by which graph?

A cubical region of side $$a$$ has its center at the origin. It encloses three fixed point charges, $$-q$$ at $$\left( {0, - a/4,0} \right), + 3q$$ at $$\left( {0,0,0} \right)$$ and $$-q$$ at $$\left( {0, + a/4,0} \right).$$ Choose the correct option(s)

A small mass m is attached to a massless string whose other end is fixed at P as shown in the figure. The mass is undergoing circular motion in the xy-plane with centre at O and constant angular speed $$\omega$$. If the angular momentum of the system, calculated about O and P are denoted by $${\overrightarrow L _O}$$ and $${\overrightarrow L _P}$$, respectively, then