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Potential Due To Continuous Charge Distribution

Practice to calculate potential due to continuous charge distribution at a point and circular ring and hollow sphere at the center, at the axis of a ring, and at the axis of a disk.

Potential due to Continuous Charge Distribution

To calculate potential due to continuous charge distribution at a point

Step 1

• Choose element such that varying the element gives us whole charge distribution.

Step 2

• Calculate charge on that element.

Step 3

• Calculate potential due to that element at a point.

Step 4

• Total potential is the scalar sum of potentials of all the elements.

Step 5

$$V= \int \dfrac {1 dq}{4\pi\epsilon_0r}$$

Case-I  Thin Rod

• Consider a thin rod of length L  and charge + Q which is equally distributed over its length.
• To calculate potential at point P, take a small element of length dx which acts as a point charge.
• Charge on the element,

$$dQ=\dfrac {Q}{L}dx$$ [ charge of element = charge per unit length × length of element ]  Case-II  Ring

• Consider a thin ring of radius R and charge + Q. To calculate potential at distance x on its axis from center, choose a small element of length $$d\ell$$, as shown in figure.
• $$d\ell$$ element acts as a point charge.
• Charge on the element,

$$dQ=\dfrac {Q}{2\pi R}d\ell$$ [ Charge of element = charge per unit length × length of element ]  Case-III   Disk

• Consider a disk of radius R and charge + Q. To calculate potential at its axial point, take a small ring as an element.
• Total potential is the scalar sum of potentials due to all the elements.
• Charge on this small ring, $$dq=\dfrac {Q}{\pi R^2}(2\pi rdr)$$

[Charge on element = Charge per unit area × Area of element]  The value of electric potential due to a disk of charge Q = +5C and radius r = 5 cm is to be calculated at an axial point(P). Choose the correct choice of element and charge on element.

A A disk of radius r = 5 cm and charge of element = $$10^3r^2\,C$$ B A ring of radius $$r\,'$$ , width dr and charge of element =  $$4\times10^3\,r'\,dr$$ C A ring of radius $$r\,'$$, width dr and charge of element = $$2\times10^3r^2\,C$$ D A rod of length $$\ell$$ at distance x from center , width dx and charge of element = $$2\times10^3\times x\; dx \,C$$ ×

Choose an element such that varying the element gives us whole charge distribution. Charge of element = charge per unit area × area of element

Charge per unit area = $$\dfrac {5}{\pi(5\times10^{-2})^2}=\dfrac {5}{\pi\times 25\times10^{-4}}$$ Area of element =  $$2\pi\,r\,'dr$$

Charge of element  =  $$\dfrac {5}{\pi\times 25\times10^{-4}}\times 2\pi r\,'dr$$

$$=4\times10^3\, r\,' dr \,C$$

The value of electric potential due to a disk of charge Q = +5C and radius r = 5 cm is to be calculated at an axial point(P). Choose the correct choice of element and charge on element.

A

A disk of radius r = 5 cm and charge of element = $$10^3r^2\,C$$

. B

A ring of radius $$r\,'$$ , width dr and charge of element =  $$4\times10^3\,r'\,dr$$ C

A ring of radius $$r\,'$$, width dr and charge of element = $$2\times10^3r^2\,C$$ D

A rod of length $$\ell$$ at distance x from center , width dx and charge of element = $$2\times10^3\times x\; dx \,C$$ Option B is Correct

Potential due to Circular Ring and Hollow Sphere at the Center

Case-I   Ring

• Consider a ring of radius R and charge +Q distributed over its length.
• To find potential at its center O, consider an element of charge dq at the circumference of ring.
• Electric potential at the center of ring due to small element of charge dq,

$$dV=\dfrac {1}{4\pi\epsilon_0}\dfrac {dq}{R}$$

• The element acts as a point charge. As all elements are at distance R from this point. So, potential due to all small elements will be same.
• Total potential at the center will be scalar sum of potentials due to all elements. So, total potential at center O,

$$V=\int \dfrac {1}{4\pi\epsilon_0}\dfrac {dq}{R}$$

$$V=\dfrac {1}{4\pi\epsilon_0}\dfrac {Q}{R}$$         $$\left[\because \;\int dq=+Q \;\right]$$  Case-II   Hollow sphere

• Consider a hollow sphere of radius R with charge +Q distributed over its surface.
• To calculate potential at its center O, choose a small element of charge dq on its surface.
• This element will act as a point charge.
• Electric potential at the center due to this element

$$dV=\dfrac {1}{4\pi\epsilon_0}\dfrac {dq}{R}$$

• Since all the elements on the surface are at equal distance from center,so potential due to all the elements will be same.
• Total potential at the center will be scalar sum of potentials of all the elements on the surface.
• Total electric potential at the center O,

$$V=\int \dfrac {1}{4\pi\epsilon_0}\dfrac {dq}{R}$$

$$V=\dfrac {1}{4\pi\epsilon_0}\dfrac {Q}{R}$$         $$\left[\because \;\int dq=+Q \;\right]$$  A hollow sphere of radius $$R=+5\,cm$$ is given a charge $$Q=5\,\mu\,C$$. Calculate the electric potential at its center. $$\left [ \dfrac {1}{4\pi\epsilon_0} =9\times 10^9\;Nm^2/C^2 \right]$$

A 0 V

B 1 × 105 V

C 2 × 105 V

D 9 × 105 V

×

Electric potential of hollow sphere of radius R and charge +Q at its center = $$\dfrac {1}{4\pi\epsilon_0} \dfrac {Q}{R}$$

$$Q=5\,\mu\,C$$ = $$5\times10^{-6}\;C$$ and $$R=5\,cm$$ = $$5\times10^{-2}\;m$$

$$V=9\times10^9\times\dfrac {5\times10^{-6}}{5\times 10^{-2}} \\=9\times10^5\;V$$

A hollow sphere of radius $$R=+5\,cm$$ is given a charge $$Q=5\,\mu\,C$$. Calculate the electric potential at its center. $$\left [ \dfrac {1}{4\pi\epsilon_0} =9\times 10^9\;Nm^2/C^2 \right]$$

A

0 V

.

B

1 × 105 V

C

2 × 105 V

D

9 × 105 V

Option D is Correct

Potential at the Axis of a Disk

• Consider a disk of radius  R  with charge  +Q distributed over its surface.  • To calculate potential at a point P at a distance  x on its axis, use the method of elements.
• Consider a small ring as an element at a distance r from center and width  dr
• Charge on an element

$$dq=\dfrac {Q}{\pi R^2}(2\pi\; r\;dr)$$

(Charge on an element = Charge per unit area × Area of element )

• Potential at point P due to this element,

$$dV=\dfrac {dq}{4\pi\epsilon_0\sqrt {r^2+x^2}}$$

Since $$dq=\dfrac {Q}{\pi R^2}(2\pi\; r\;dr)$$

$$dV=\dfrac {Q\; (2\pi\; r\;dr)}{\pi R^2\times 4\pi\epsilon_0\sqrt {r^2+x^2}}$$

$$dV=\dfrac {Q.\; r\;dr}{2\pi\epsilon_0 R^2\sqrt {r^2+x^2}}$$  • ?Integrating the potential from  r = 0  to  r = R  gives the total potential of whole disk,

$$V=\int\limits_0^R\dfrac {Q}{2\pi\epsilon_0R^2} \dfrac {rdr}{\sqrt{r^2+x^2}}$$

$$V=\dfrac {Q}{2\pi\epsilon_0R^2} \int\limits_0^R \dfrac {rdr}{\sqrt{r^2+x^2}}$$

$$Put\,\,\,\,{r^2 + x^2}={t^2}$$

By partial differentiation,  $${ 2r\,+0}=2t\,dt$$

$$\rightarrow r\,dr=t\,dt$$

$$Limits\rightarrow\,\,\,\,{r^2 + x^2}={t^2}$$

$$at\,r=0\,\,\,\,\,\,{t^2}={x^2} \\ \,\,\,\,\,\,\,\,\,\,\,t=x \\at\,r=R\,\,\,\,\,{R^2+x^2}={t^2} \\\,\,\,\,\,\,\,\,\,\,t=\sqrt{R^2+x^2}$$

So, $$V=\dfrac {Q}{2\pi\epsilon_0R^2} \int\limits_x^\sqrt{R^2+x^2} \dfrac {t\; dt}{t}$$

$$V=\dfrac {Q}{2\pi\epsilon_0R^2} [\;t\;]_x^\sqrt{R^2+x^2}$$

$$V=\dfrac {Q}{2\pi\epsilon_0R^2} [\sqrt{R^2+x^2}-x]$$

Choose the option for correct sequence of steps for obtaining potential due to a ring at a point on its axis.

A B C D

×

To calculate potential due to continuous charge distribution at a point.

Choose element such that varying the element gives us whole charge distribution.

Calculate charge on that element.

Calculate potential due to that element at a point.

Total potential is the scalar sum of potentials of all the elements.

$$V= \int \dfrac {1 dq}{4\pi\epsilon_0r}$$

Choose the option for correct sequence of steps for obtaining potential due to a ring at a point on its axis.

A B C D

Option B is Correct

Potential at any point due to combinations

Steps to calculate potential at any point due to combinations

Step 1

Calculate potential due to different-different charge distribution.

Step 2

Total potential is the scalar sum of potentials due to all charge distributions.

Calculate electric potential at point  $$P$$ at a distance $$x=4\,cm$$ from the center on the common axis of two concentric circular objects. The inner one is a disk of radius $$R_D=3\,cm$$  and charge $$Q_D=-5\,C$$ and outer one is the ring of radius $$R_R=6\,cm$$  and charge $$Q_R=10\sqrt{13}\,C$$ .$$\left [ \dfrac {1}{4\pi\epsilon_0}=9\times10^9\; Nm^2/C^2 \right]$$

A 3.5 × 1012 V

B 9.5 × 1015 V

C 9.2 × 108 V

D 1.1 × 106 V

×

Potential at point P due to the disk,

x = 4 cm, RD = 3 cm and QD = –5 C

$$V_{Disk}= \dfrac {Q}{2\pi\epsilon_0R^2} [\sqrt{R^2+x^2}-x]$$

$$V_{Disk}= \dfrac {(-5)\times2\times 9\times10^9}{(3\times10^{-2})^2} [\sqrt{(3\times10^{-2})^2+(4\times10^{-2})^2}-(4\times10^{-2})]$$

$$V_{Disk}=-10^{12}\; Volt$$ Potential at point P due to the ring,

$$x=4\,cm$$ , $$R_R=6\,cm$$  and $$Q_R=10\sqrt{13}\,C$$

$$V_{Ring}= \dfrac {1}{4\pi\epsilon_0} \times \dfrac {Q}{\sqrt{R^2+x^2}}$$

$$V_{Ring}= \dfrac {9\times10^9\times 10\sqrt{13}}{\sqrt {(6)^2+(4)^2}}$$

$$V_{Ring}= \dfrac {9\times10^9\times 10\sqrt{13}}{2\sqrt {13}\times10^{-2}}$$

$$V_{Ring}=45\times10^{11}$$  $$V_{Ring}=4.5\times10^{12}\; Volt$$ Total electric potential at point P,

$$V=V_{Disk}+V_{Ring}$$

$$V=(-10^{12})+4.5\times10^{12}$$

$$V=3.5\times10^{12}\; Volt$$

Calculate electric potential at point  $$P$$ at a distance $$x=4\,cm$$ from the center on the common axis of two concentric circular objects. The inner one is a disk of radius $$R_D=3\,cm$$  and charge $$Q_D=-5\,C$$ and outer one is the ring of radius $$R_R=6\,cm$$  and charge $$Q_R=10\sqrt{13}\,C$$ .$$\left [ \dfrac {1}{4\pi\epsilon_0}=9\times10^9\; Nm^2/C^2 \right]$$ A

3.5 × 1012 V

.

B

9.5 × 1015 V

C

9.2 × 108 V

D

1.1 × 106 V

Option A is Correct

Potential at the Axis of a Ring

• Ring can be considered as combination of point charges and total potential at point P is the scalar sum of potentials due to all charges.
• Total potential at point P,
• $$V=\int\dfrac {dq}{4\pi\epsilon_0\sqrt {R^2+x^2}}$$

$$V=\dfrac {q}{4\pi\epsilon_0\sqrt {R^2+x^2}}$$  Choose the correct option of steps for obtaining potential due to a ring at a point on its axis.

A B C D

×

To calculate potential due to continuous charge distribution at a point.

Choose element such that varying the element gives us whole charge distribution.

Calculate charge on that element.

Calculate potential due to that element at a point.

Total potential is the scalar sum of potentials of all the elements.

$$V= \int \dfrac { dq}{4\pi\epsilon_0r}$$

Choose the correct option of steps for obtaining potential due to a ring at a point on its axis.

A B C D

Option B is Correct

Potential at the Axis of  a Charged Rod

• Consider a thin rod of length L and charge +Q .
• To calculate electric potential at a point P on its axis situated at a distance r from the center of the rod, use method of element.
• Choose an element at a distance  x  from center of rod of length dx.
• This element acts as point charge.
• Charge on this element,

$$dq=\dfrac {Q}{L}\times dx$$ [ Charge on element = charge per unit length × length of the element]

• To calculate electric potential at point P due to this element, first calculate its distance from point P. This distance will be (r–x).
• Electric potential at point P due to this element,

$$dV=\dfrac {1}{4\pi\epsilon_0}\dfrac {dq}{(r-x)}$$

• Electric potential is a scalar quantity.
• Total potential at point P is the scalar sum of potentials due to all small elements.
• Total electric potential

$$V=\int \dfrac {1}{4\pi\epsilon_0}\dfrac {dq}{(r-x)}$$

$$V=\int\limits_{-L/2}^{L/2} \dfrac {1}{4\pi\epsilon_0} \;\dfrac {Qdx}{L(r-x)}$$

$$V=\dfrac {1}{4\pi\epsilon_0} \;\dfrac {Q}{L} \int\limits_{-L/2}^{L/2} \;\dfrac {dx}{r-x}$$

$$V=\dfrac {1}{4\pi\epsilon_0} \;\dfrac {Q}{L} [-\ell n(r-x)]_{-L/2}^{L/2}$$

$$\because$$ $$\int\limits_m^n \left( \;\dfrac {1}{a-x} \right) dx=[-\ell n(a-x)]_{m}^{n}$$

$$V= \dfrac {1}{4\pi\epsilon_0} \dfrac {Q}{L} \left[ -\ell n \left (r-\dfrac {L}{2}\right)+\ell n \left (r+\dfrac {L}{2}\right) \right]$$

$$V= \dfrac {1}{4\pi\epsilon_0} \dfrac {Q}{L} \ell n \left( \dfrac {r+L/2}{r-L/2} \right)$$  Choose the correct sequence of steps to calculate electric potential due to charged rod at its axis.

A B C D

×

To calculate potential due to continuous charge distribution at a point.

Choose element such that varying the element gives us whole charge distribution.

Calculate charge on that element.

Calculate potential due to that element at a point.

Total potential is the scalar sum of potentials of all the elements.

$$V= \int \dfrac {dq}{4\pi\epsilon_0r}$$

Choose the correct sequence of steps to calculate electric potential due to charged rod at its axis.

A B C D

Option C is Correct