The electrostatic potential in a charged spherical region of radius $r$ varies as $V = a r^{3} + b,$ where $a$ and $b$ are constants. The total charge in the sphere of unit radius is $α \times π a ε_{0}$ . The value of $α$ is _________.

(Permittivity of vacuum is $ε_{0}$ ) [2026]

Question

The electrostatic potential in a charged spherical region of radius $r$ varies as $V = a r^{3} + b,$ where $a$ and $b$ are constants. The total charge in the sphere of unit radius is $α \times π a ε_{0}$ . The value of $α$ is _________.

(Permittivity of vacuum is $ε_{0}$ ) [2026]

Smart Achievers · Accepted Answer

(1)

$V = a r^{3} + b$

$E = - \frac{d V}{d r} = - 3 a r^{2}$

$ϕ_{closed} = \frac{q_{enc}}{ε_{0}}$

$q_{enc} = ε_{0} \cdot E \cdot A$

$= ε_{0} (- 3 a \cdot {(1)}^{2}) 4 π {(1)}^{2}$

$= - 12 π a ε_{0}$

$∴$ $x = - 12$

Solution

Q.
The electrostatic potential in a charged spherical region of radius $r$ varies as $V = a r^{3} + b,$ where $a$ and $b$ are constants. The total charge in the sphere of unit radius is $α \times π a ε_{0}$ . The value of $α$ is _________.

(Permittivity of vacuum is $ε_{0}$ ) [2026]

1 − 12

2 − 8

3 − 9

4 − 6

Ans.
(1)

$V = a r^{3} + b$

$E = - \frac{d V}{d r} = - 3 a r^{2}$

$ϕ_{closed} = \frac{q_{enc}}{ε_{0}}$

$q_{enc} = ε_{0} \cdot E \cdot A$

$= ε_{0} (- 3 a \cdot {(1)}^{2}) 4 π {(1)}^{2}$

$= - 12 π a ε_{0}$

$∴$ $x = - 12$

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