Tag: relation between electric field and electric potential

Questions Related to relation between electric field and electric potential

In a certain region of space, the potential is given by $V=k\left[ { 2x }^{ 2 }-{ y }^{ 2 }+{ z }^{ 2 } \right] $. The electric field at the point$ (1,1,1)$ has magnitude :

physics coulomb's law field strength and potential gradient electric field as gradient of potential relation between electric field and electric potential

  1. $k\sqrt { 6 } $

  2. $2k\sqrt { 6 } $

  3. $2k\sqrt { 3 } $

  4. $4k\sqrt { 3 } $

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Correct Option: B
Explanation:

Given, $V=k[2x^2-y^2+z^2]$

Electric field , $\vec{E}=-(\dfrac{dV}{dx}\hat i+\dfrac{dV}{dy}\hat j+\dfrac{dV}{dz}\hat {k})$

or,$\vec{E}=-k(4x\hat i-2y\hat j+2z\hat k)$

or,$\vec{E} _{(1,1,1)}=-k(4\hat i-2\hat j+2\hat k)$

Magnitude of electric field$ =|\vec{E} _{(1,1,1)}|=\sqrt{k^2(16+4+4)}=k\sqrt {24}=2k\sqrt 6$

Let V be electric potential and E the magnitude of the electric field. At a given position, which of the statement is true

physics coulomb's law field strength and potential gradient electric field as gradient of potential relation between electric field and electric potential

  1. E is always zero where V is zero

  2. V is always zero where E is zero

  3. E can b zero where V is non zero

  4. E is always nonzero where V is nonzero

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Correct Option: A

Two plates are 1 cm apart and the potential difference between them is 10 volt. The electric field between the plates is

physics coulomb's law field strength and potential gradient electric field as gradient of potential relation between electric field and electric potential

  1. 10 N/C

  2. 250 N/C

  3. 500 N/C

  4. 1000 N/C

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Correct Option: D

The equation of an equipotential line in an electric field is $y=2x$, then the electric field strength vector at $(1,2)$ may be :

physics coulomb's law field strength and potential gradient electric field as gradient of potential relation between electric field and electric potential

  1. $4\hat { i } +3\hat { j } $

  2. $4\hat { i } +8\hat { j } $

  3. $8\hat { i } +4\hat { j } $

  4. $-8\hat { i } +4\hat { j } $

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Correct Option: D
Explanation:

Now equation of equipotential surface is $y=2x$
Now electric field along the euipotential surface should be zero
therefore angle made by equipotential surface with x-axis is $tan^{-1} { (2) } $
Now since net electric field should be perpendicular to the equipotential surface
therefore for any electric field which makes an angle $tan^{-1} { (-1/2) } $ with x-axis can be the electric field at point $(1,2)$ which is true only for option (D)

because for two perpendicular line, product of their slope should be equal to -1 i.e., $m _1 \times m _2=-1$

Two plates are at potentials $-10 V$ and $+30 V$. If the separation between the plates is $2 cm$ then the electric field between them will be 

physics coulomb's law field strength and potential gradient electric field as gradient of potential relation between electric field and electric potential

  1. 2000 V/m

  2. 1000 V/m

  3. 500 V/m

  4. 3000 V/m

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Correct Option: A
Explanation:

Given,

$d=2cm$
$V _2-V _1=+30V-(-10)=40V$
Electric field, $E=\dfrac{V _2-V _1}{d}$
$E=\dfrac{40}{2\times 10^{-2}}=2000V/m$
The correct option is A.

In a certain region the electric potential at a point $(x, y, z)$ is given by the potential function $V = 2x + 3y - z$. Then the electric field in this region will :

physics coulomb's law field strength and potential gradient electric field as gradient of potential relation between electric field and electric potential

  1. increase with increase in x and y

  2. increase with increase in y and z

  3. increase with increase in z and x

  4. remain constant

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Correct Option: D
Explanation:

$V=2x+3y-z$

$E _x=-\dfrac{dV}{dx}=-2,  E _y=-\dfrac{dV}{dy}=-3 $ and $E _z=-\dfrac{dV}{dz}=1$

As the field components are independent of x,y and z so the field remains constant.

Which of the following is true for uniform electric field ?

physics coulomb's law field strength and potential gradient electric field as gradient of potential relation between electric field and electric potential

  1. all points are at the same potential

  2. no two points can have the same potential

  3. pairs of points separated by the different distance must have the same difference in potential

  4. none of the above

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Correct Option: D
Explanation:

uniform electric field means the electric field vector does not vary with positive and electric field lines are parallel and equally spaced. the statement given define the equipotential surface, so answer is (d) -none of these.

The electric field and the electric potential at a point are E and V respectively. Then, the incorrect statements are :

physics coulomb's law field strength and potential gradient electric field as gradient of potential relation between electric field and electric potential

  1. If E $=$ 0, V must be zero.

  2. If V $=$ 0, E must be zero.

  3. If E $\neq$ 0, V cannot be zero.

  4. If V $\neq$ 0, E cannot be zero.

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Correct Option: A,B,C,D
Explanation:

The electric field is $E=-\dfrac{dV}{dx}$
If $V=0$, we can not say $E$ must be zero, we say only $E$ may be zero.
If $V \neq 0 $, $E$ must be zero when $V$ is max i.e, $\dfrac{dV}{dx}=0$ For example, inside the conductor $E=0,$ but $V \neq 0 $
If $E \neq 0$ , $V$  may be zero when two equal and opposite charges separated by a distance and  at the midpoint in between the charges field is non-zero but potential is zero.

A charge of $6.76$ $\mu$C in an electric field is acted upon by a force of $2.5 N$. The potential gradient at this point is :

physics coulomb's law field strength and potential gradient electric field as gradient of potential relation between electric field and electric potential

  1. $3.71 \times 10^{15} Vm^{-1}$

  2. $-3.71 \times 10^{12} Vm^{-1}$

  3. $3.71 \times 10^{10} Vm^{-1}$

  4. $-3.71 \times 10^{5} Vm^{-1}$

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Correct Option: D
Explanation:

$F=qE \Rightarrow E=\dfrac{F}{q}=\dfrac{2.5}{6.76\times 10^{-6}}=3.7\times 10^5$

The potential gradient is the electric field i.e, $E=-\nabla V=-3.7 \times 10^5 $

The electric potential decreases uniformly from $120$ V to $80$ V as one moves on the X-axis from x $=$ -1 cm to x $=$ $+1$ cm. The electric field at the origin :

physics coulomb's law field strength and potential gradient electric field as gradient of potential relation between electric field and electric potential

  1. must be equal to $20$ V/cm

  2. may be equal to $20$ V/cm

  3. may be greater than $20$ V/m

  4. may be less than $20$ V/cm

Show answer
Correct Option: A
Explanation:

As the electric potential decreases uniformly so the electric field is uniform along the x axis.

$E=-\dfrac{dV}{dx}=\dfrac{120-80}{1-(-1)}=20 $V/m. It must be equal to 20 V/m at the origin.