(C) The electric field $E$ at a distance $r$ from an infinitely long charged wire is given by $E = \frac{2K\lambda}{r}$.The work done by an external a

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(C) The electric field $E$ at a distance $r$ from an infinitely long charged wire is given by $E = \frac{2K\lambda}{r}$.The work done by an external agent to move a charge $q$ is $W_{ext} = \Delta U = -\int_{a}^{2a} \vec{F} \cdot d\vec{r} = -\int_{a}^{2a} q\vec{E} \cdot d\vec{r}$.Substituting the expression for $E$: $W_{ext} = -\int_{a}^{2a} q \left( \frac{2K\lambda}{r} \right) dr$.$W_{ext} = -2K\lambda q \int_{a}^{2a} \frac{1}{r} dr$.$W_{ext} = -2K\lambda q [\ln(r)]_{a}^{2a}$.$W_{ext} = -2K\lambda q (\ln(2a) - \ln(a)) = -2K\lambda q \ln\left(\frac{2a}{a}\right)$.$W_{ext} = -2K\lambda q \ln(2)$.

Class 12 Physics Electric Charges and Fields Electric Field

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An infinitely long uniformly charged string is placed along the $z-$axis. Its linear charge density is $\lambda$. $A$ point charge $q$ is moved from position $(a, 0, 0)$ to $(2a, 0, 0)$. The work done by the external agent is:

A
$-\frac{K\lambda q}{a} \ln(2)$
B
$\frac{K\lambda q}{2a} \ln(2)$
C
$-2K\lambda q \ln(2)$
D
$- \frac{2K\lambda q}{a} \ln(2)$

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Electric Charges and Fields

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Electric Field

Three isolated metal spheres $A$,$B$,and $C$ have radii $R$,$2R$,and $3R$ respectively,and the same charge $Q$. If $U_A$,$U_B$,and $U_C$ are the energy densities just outside the surfaces of the spheres,then the relation between $U_A$,$U_B$,and $U_C$ is:

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$A$ negative charge $-q$ is placed at the midpoint between two fixed equal positive charges $Q$, separated by a distance $2d$. If the negative charge is given a small displacement $x$ $(x \ll d)$ perpendicular to the line joining the positive charges, how will the net force $(F)$ acting on it approximately depend on $x$?

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Four point positive charges of same magnitude $Q$ are placed at four corners of a rigid square frame in $xy$ plane as shown in the figure. The plane of the frame is perpendicular to the $z$-axis. If a negative point charge is placed at a distance $z$ away from the above frame $(z << L)$,then:

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$A$ conducting sphere of radius $10 \, cm$ has an unknown charge. If the electric field at a distance $20 \, cm$ from the centre of the sphere is $1.2 \times 10^3 \, N \, C^{-1}$ and points radially inwards,the net charge on the sphere is:

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$A$ thin glass rod is bent in a semicircle of radius $R$. $A$ charge is non-uniformly distributed along the rod with a linear charge density $\lambda = \lambda_0 \sin \theta$ (where $\lambda_0$ is a positive constant and $\theta$ is the angle with the $x$-axis). The electric field at the centre $P$ of the semicircle is,

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An infinitely long uniformly charged string is placed along the $z-$axis. Its linear charge density is $\lambda$. $A$ point charge $q$ is moved from position $(a, 0, 0)$ to $(2a, 0, 0)$. The work done by the external agent is:

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Three isolated metal spheres $A$,$B$,and $C$ have radii $R$,$2R$,and $3R$ respectively,and the same charge $Q$. If $U_A$,$U_B$,and $U_C$ are the energy densities just outside the surfaces of the spheres,then the relation between $U_A$,$U_B$,and $U_C$ is:

Four point positive charges of same magnitude $Q$ are placed at four corners of a rigid square frame in $xy$ plane as shown in the figure. The plane of the frame is perpendicular to the $z$-axis. If a negative point charge is placed at a distance $z$ away from the above frame $(z << L)$,then:

$A$ conducting sphere of radius $10 \, cm$ has an unknown charge. If the electric field at a distance $20 \, cm$ from the centre of the sphere is $1.2 \times 10^3 \, N \, C^{-1}$ and points radially inwards,the net charge on the sphere is:

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