Potential Gradient and Relation between Electric Field and Potential

Q. The electric potential V at any point O (x, y, z all in metres) in space is given by $V=4x^2$ volt. The electric field at the point (1m, 0, 2m) in volt/metre is

1. 8 along negative X- axis
2. 8 along positive X - axis
3. 16 along negative X - axis
4. 16 along positive Z - axis
   

Q. In a region, the potential is represented by $V(x,y,z)=6x-8xy+6yz$, where $V$ is in volts and $x$, $y$ and $z$ are in metres. The electric force experienced by a charge of $2$ Coulomb situated at point $(1,1,1)$ is

1. $6\sqrt{5}N$
2. $30N$
3. $24N$
4. $4\sqrt{35}N$

Q. An electric field is expressed as $\overrightarrow{E}=2\hat{i}+3\hat{j}$. Find the potential difference $(V_A-V_B)$ between two points A and B whose position vectors are given by $r_A=\hat{i}+2\hat{j}$ and $r_B=2\hat{i}+\hat{j}+3\hat{k}.$

1. $-1V$
2. $+1V$
3. $2V$
4. $3V$

Q. The electric potential at a point (x, y) in the x - y plane is given by V = - kxy. The field intensity at a distance r from the origin varies as

1. $r^2$
2. $r$
3. $\frac{1}{r}$
4. $\frac{1}{r^2}$

Q. The electric field intensity at points $P$ and $Q$ in the shown arrangement, are in the ratio

1. $2:1$
2. $4:3$
3. $1:2$
4. $1:1$

Q. Find $\left(a\right)V_B-V_A\left(b\right)V_C-V_B\left(c\right)V_A-V_C$ for the following uniform E-field

1. -1V, -1V, +2V
2. -1V, 3V, +2V
3. -1V, -3V, +3V
4. -1V, 1V, +2V

Q. A solid conducting sphere having a charge $Q$ is surrounded by an uncharged concentric conducting hollow spherical shell. Let the potential difference between the surface of the solid sphere and that of the outer surface of the hollow shell be $V$. If the shell is now given a charge $-4Q$, the new potential difference between the same two surfaces is :

1. 4 V
2. 2 V
3. -2 V
4. V

Q. Two points are at a distance $a$ and $b$ apart $(a<b)$ from left end of a uniformly charged rod as shown in the figure. The difference between the potentials at the two given points is proportional to

1. $ln\left(\frac{b}{a}\right)$
2. $\frac{b}{a}$
3. $\sqrt{\frac{b}{a}}$
4. $$ln\left\{\frac{b(a-L)}{a(b-L)}\right\}$$

Q. Electric potential is given by $V=6x-8x{y}^2-8y+6yz-4z^2$
Then electric force acting on a 2C point charge placed at origin will be?

1. 2 N
2. 6 N
3. 8 N
4. 20 N

Q. The potential field of an electric field $\overrightarrow{E}=(y\hat{i}+x\hat{j})$ is

1. $V=-xy+\text{constant}$
2. $V=xy+\text{constant}$
3. $V=-(x^2+y^2)+\text{constant}$
4. $V=\text{constant}$

Q. A non-conducting sphere of radius $0.5\text{m}$ carries a total charge of $10\times {10}^{-10}\text{C}$ distributed uniformly which produces an electric field. Then the value of integral
$-\underset{\infty }{\overset{0}{\int }}\overrightarrow{E}\cdot \overrightarrow{dr}$ will be

1. $27$
2. $0$
3. $-27$
4. $-9$

Q. A uniform electric field of magnitude $E_0$ is directed along the positive X - axis. If the potential V is zero at x = 0, then its value at X = + x will be

1. $V_x=+x{E}_0$
2. $V_x=-x{E}_0$
3. $V_x=+x^2{E}_0$
4. $V_x=-x^2{E}_0$

Q. If potential (in volts) in a region is expressed as $V(x,y,z)=6xy-y+2yz$, the electric field (in $N/C$) at point $(1,1,0)$is

1. $-\left(6\hat{i}+9\hat{j}+\hat{k}\right)$
2. $-\left(3\hat{i}+5\hat{j}+3\hat{k}\right)$
3. $-\left(6\hat{i}+5\hat{j}+2\hat{k}\right)$
4. $-\left(2\hat{i}+3\hat{j}+\hat{k}\right)$

Q. Figure shows two equipotential lines in $\text{XY}$ plane for an electric field. The scales are marked. The $\text{X-}$component $E_x$ and $\text{Y-}$component $E_y$ of the electric field in the space between these equipotential lines are respectively.

1. $-100{\text{Vm}}^{-1}$, $+200{\text{Vm}}^{-1}$
2. $-200{\text{Vm}}^{-1}$, $+100{\text{Vm}}^{-1}$
3. $+200{\text{Vm}}^{-1}$, $+100{\text{Vm}}^{-1}$
4. $+100{\text{Vm}}^{-1}$, $-200{\text{Vm}}^{-1}$

Q. In the following figure two parallel metallic plates are maintained at different potentials. If an electron is released midway between the plates, it will move

1. Right ward at constant speed
2. Left ward at constant speed
3. Accelerates right ward
4. Accelerates left ward

Q. If potential in a region of space is given by $V=x^{2}y$, what is the electric field in the region?

1. $\hat{i}+\hat{j}+\hat{k}$
2. $-2xy\hat{i}-x^2\hat{j}$
3. $-x^2\hat{i}-2xy\hat{j}$
4. None of these

Q. If the electric potential along a line due to a charge is given by $V\left(r\right)=\frac{25}{r}+3$, where r is the distance from the charge, what is the electric field at a point r = 5 along this line?

1. $-1\frac{N}{C}$
2. $1\frac{N}{C}$
3. $5\frac{N}{C}$
4. $-5\frac{N}{C}$

Q. Find the potential $V$ of an electrostatic field $\overrightarrow{E}=a(y\hat{i}+x\hat{j}),$ where a is a constant.

1. $axy+C$
2. $-axy+C$
3. $axy$
4. $-axy$

Q. An electric field is given by $\overrightarrow{E}=(y\hat{i}+x\hat{j})\text{N/C}$. The work done in moving a $1\text{C}$ charge from $\overrightarrow{r_A}=(2\hat{i}+2\hat{j})\text{m}$ to $\overrightarrow{r_B}=(4\hat{i}+\hat{j})\text{m}$ is

1. $+4\text{J}$
2. $-4\text{J}$
3. $+8\text{J}$
4. Zero

Q. $\text{ABC}$ is an equilateral triangle of side $2\text{m}$. If electric field $E=10{\text{NC}}^{-1}$ then the potential difference $V_A-V_B$ is

1. $-10\text{V}$
2. $10\text{V}$
3. $-20\text{V}$
4. $20\text{V}$

Q. The electric potential varies in space according to the relation $V=3x+4y.$ A particle of mass $0.1kg$ starts from rest from point $(2,3.2)$ under the influence of this field. The charge on the particle is $+1\mu C.$ Assume V and $(x,y)$ are in S.I. units
The component of electric field in the y - direction $\left(E_y\right)$ is

1. $3V{m}^{-1}$
2. $-4V{m}^{-1}$
3. $5V{m}^{-1}$
4. $8V{m}^{-1}$

Q.

The potential at a point x (measured in μm) due to some charges situated on the x-axis is given by
V(x) = 20 / $(x^2$ - 4) Volt
The electric field E at x = 4 μm is given by:
[AIEEE-2007]

1. $\frac{5}{3}$ Volt/μm and in the positive x direction

2. $\frac{10}{9}$ Volt/μm and in the negative x-direction

3. $\frac{10}{9}$ Volt/μm and in the positive x-direction

4. $\frac{5}{3}$ Volt/μm and in the negative direction

Q. If $\overrightarrow{E}=E_0(y\hat{i}+x\hat{j})$ find the potential difference between (-1, 3)and (2, 4)

1. \N
2. $11E_0$
   
3. $-11E_0$
   
4. $5E_0$

Q. Electric field is space is given by $\overrightarrow{E}=\frac{E_0}{a}x\hat{j}(E_0=10N/C$, a = 1 metre), find potential of point (4, 3) (assuming potential at origin is 100 volt)

1. 10 volt
2. 25 volt
3. 20 volt
4. 5 volt

Q. If points A and B have a potential 3V and 5V respectively, and a dipole is brought between the two points, which way will its dipole moment align?
Given: Dipole is an arrangement of two equal and opposite charges separated by a very small fixed distance and Dipole moment is a vector quantity which points from negative to the positive charge of the dipole arrangement.

1. Towards A
2. Towards B
3. Neither A nor B
4. Can’t say

Q. Find the potential $V$ of an electrostatic field $\overrightarrow{E}=a(y\hat{i}+x\hat{j}),$ where a is a constant.

1. $axy+C$
2. $-axy+C$
3. $axy$
4. $-axy$

Q. In moving from A to B along an electric field line, the electric field does $6.4\times {10}^{-19}J$ of work on an electron. If ${\varphi }_1,{\varphi }_2$ are equipotential surfaces, then the potential difference $(V_C-V_A)$ is?

1. -4V
2. 4V
3. Zero
4. 64V

Q. A uniform electric field pointing in positive x-direction exists in a region. Let A be the origin, B be the point on the x-axis at x = +1 cm and C be the point on the y-axis at y = +1 cm. Then the potentials at the points A, B and C satisfy

1. $V_A<V_B$
2. $V_A>V_B$
3. $V_A<V_C$
4. $V_A>V_C$

Q. A small sphere with mass $1.2g$ hangs by a thread between two parallel vertical plates $5cm$ apart. The plates are insulating and have uniform surface charge densities $+\delta$ and $-\delta$ . The charge on the sphere is $q=9\times {10}^{-6}C$. What potential difference (in V) between the plates will cause the thread to assume an angle of ${37}^{\circ }$ with the vertical as shown in figure?

Q. In moving from A to B along an electric field line, the electric field does $6.4\times {10}^{-19}J$ of work on an electron. If ${\varphi }_1,{\varphi }_2$ are equipotential surfaces, then the potential difference $(V_C-V_A)$ is?

1. -4V
2. 4V
3. Zero
4. 64V