Derive an expression for electric field due to an electric dipole at a point on its axial line.

Electric field due to an electric dipole at a point on its axial line: \\(AB\\) is an electric dipole of two point charges \\(-q\\) and \\(+q\\) separated by small distance \\(2d\\). \\(P\\) is a point along the axial line of the dipole at a distance \\(r\\) from the midpoint \\(O\\) of the electric dipole.The electric fiedl at the point \\(P\\) due to \\(+q\\) placed at \\(B\\) is,\\({ E }_{ 1 }=\dfrac { 1 }{ 4\pi { \varepsilon }_{ 0 } } \dfrac { q }{ { \left( r-d \right) }^{ 2 } }\\) (along \\(BP\\))The electric field at the point \\(P\\) due to \\(-q\\) placed at \\(A\\) is,\\({ E }_{ 2 }=\dfrac { 1 }{ 4\pi { \varepsilon }_{ 0 } } \dfrac { q }{ { \left( r+d \right) }^{ 2 } }\\) (along \\(PA\\))Therefore, the magnitude of resultant electric field \\(\left( E \right)\\) acts in the direction of the vector with a greater, magnitude. The resultant electric field at \\(P\\) is\\(E={ E }_{ 1 }+\left( -{ E }_{ 2 } \right)\\)\\(E=\left[ \dfrac { 1 }{ 4\pi { \varepsilon }_{ 0 } } \dfrac { q }{ { \left( r-d \right) }^{ 2 } } - \dfrac { 1 }{ 4\pi { \varepsilon }_{ 0 } } \dfrac { q }{ { \left( r+d \right) }^{ 2 } } \right]\\) along \\(BP\\)\\(E=\dfrac { q }{ 4\pi { \varepsilon }_{ 0 } } \left[ \dfrac { 1 }{ { \left( r-d \right) }^{ 2 } } - \dfrac { 1 }{ { \left( r+d \right) }^{ 2 } } \right]\\) along \\(BP\\)\\(E=\dfrac { q }{ 4\pi { \varepsilon }_{ 0 } } \left[ \dfrac { 4rd }{ { \left( { r }^{ 2 }-{ d }^{ 2 } \right) }^{ 2 } } \right]\\) along \\(BP\\)If the point \\(P\\) is far away from the dipole, then \\(d \ll r\\)\\(\therefore E=\dfrac { q }{ 4\pi { \varepsilon }_{ 0 } } - \dfrac { 4rd }{ { r }^{ 4 } } = \dfrac { q }{ 4\pi { \varepsilon }_{ 0 } } \dfrac { 4d }{ { r }^{ 3 } }\\)\\(E=\dfrac { 1 }{ 4\pi { \varepsilon }_{ 0 } } \dfrac { 2p }{ { r }^{ 3 } }\\) along \\(BP\\) [\\(\because \\) Electric dipole moment \\(p=q\times 2d\\)]\\(E\\) acts in the direction of dipole moment.

Derive an expression for electric potential at a point due to an electric dipole. Discuss the special cases.

Let an electric dipole consist of two equal and opposite point charges \\(– q\\) at A and \\(+q\\) at B ,separated by a small distance AB =\\(2a\\) ,with centre at O.The dipole moment, \\(p=q\times2a\\)We will calculate potential at any point P, where\\(OP=r\\) and \\(\angle BOP= \theta\\)Let \\(BP=r_1\\) and \\(AP=r_2\\)Draw AC perpendicular PQ and BD perpendicular POIn \\(\Delta AOC\\) \\(cos\theta=OC/OA =OC/a\\)\\(OC= a cos \theta\\)Similarly, \\( OD= a cos \theta\\)Potential at P due to \\(+q=\dfrac{1}{4\pi\epsilon_0}\dfrac{q}{r_2}\\)And Potential at P due to \\(-q=\dfrac{1}{4\pi\epsilon_0}\dfrac{q}{r_1}\\)Net potential at P due to the dipole\\(V=\dfrac{1}{4\pi\epsilon_0}(\dfrac{q}{r_2}-\dfrac{q}{r_1})\\)\\(\implies V=\dfrac{q}{4\pi\epsilon_0}(\dfrac{1}{r_2}-\dfrac{1}{r_1})\\)Now, \\(r_1=AP=CP\\)\\(=OP+OC\\)\\(=r+a cos\theta\\)And \\(r_2=BP=DP\\)\\(=OP –OD\\)\\(=r-acos\theta\\)\\(V=\dfrac{q}{4\pi\epsilon_0}(\dfrac{1}{r-acos\theta}-\dfrac{1}{r+acos\theta})\\)\\(=\dfrac{q}{4\pi\epsilon_0}(\dfrac{2acos\theta}{r^2-a^2cos^2\theta})\\)\\(=\dfrac{pcos\theta}{r^2-a^2cos^2\theta}\\)(Since \\(p=2aq\\))Special cases:-(i) When the point P lies on the axial line of the dipole, \\(\theta=0^{\circ}\\)\\(cos\theta=1\\)\\(V=\dfrac{p}{r^2-a^2}\\)If \\(a<<r\\), \\(V=\dfrac{p}{r^2}\\)Thus due to an electric dipole ,potential, \\(V\propto \dfrac{1}{r^2}\\)(ii) When the point P lies on the equatorial line of the dipole, \\(\theta=90^{\circ}\\)\\(cos\theta=0\\)i.e electric potential due to an electric dipole is zero at every point on the equatorial line of the dipole.

Derive an expression for electric field due electric dipole at a point on an equatorial line.

Electric field at A due to the charges \\(q\\) and \\(-q\\) is shown in the above figure. We resolve \\(E\\) into horizontal and vertical components. The vertical components \\((E \sin\theta)\\) cancel out each other and only the horizontal components survive to give net electric field at A as \\(2E\cos\theta\\).Electric field at point A, \\(E_A = 2E \cos\theta\\)where \\(E = \dfrac{q}{4\pi \epsilon_o (r^2 + a^2)}\\)We get \\(E_A = \dfrac{2q }{4\pi \epsilon_o (r^2 + a^2)} \cos\theta\\)From figure, we have \\(\cos\theta = \dfrac{a}{\sqrt{r^2 + a^2}}\\)\\(\therefore\\) \\(E_A = \dfrac{2q a}{4\pi \epsilon_o (r^2 + a^2)^{3/2}} \\)We know dipole moment \\(P = 2qa\\)\\(\implies\\) \\(E_A = \dfrac{P}{4\pi \epsilon_o (r^2 + a^2)^{3/2}} \\)For \\(a<<r\\), we can neglect \\(a^2 \\) compared to \\(r^2\\).\\(\implies\\) \\(E_A = \dfrac{P}{4\pi \epsilon_o r^3} \\)

Derive an expression for the electric field at a point on the equatorial plane of an electric dipole.

Consider an electric dipole AB consisting charges +q and -q separated by distance 2l and pole strength q. Let, P be the point on the equatorial line at distance d from mid-point of the dipole i.e. O as shown in figure 1. Now, the electric field, \\(E_1\\) at point P due to north pole along AP is, \\(E_1 = \dfrac{1}{4\pi \epsilon_0} \dfrac{q}{AP^2}\\) \\(E_1 = \dfrac{1}{4\pi \epsilon_0} \dfrac{q}{(d^2 + l^2)^2}\\) .........(1) (\\(\because AP^2 = OA^2 + OP^2\\)) Similarly, the electric field, \\(E_2\\) at point P due to south pole along PB is, \\(E_2 = \dfrac{1}{4\pi \epsilon_0} \dfrac{q}{PB^2}\\)\\(E_2 = \dfrac{1}{4\pi \epsilon_0} \dfrac{q}{(d^2 + l^2)^2}\\) .........(2) (\\(\because PB^2 = OB^2 + OP^2\\))From equations (1) and (2), we get\\(E_1 = E_2 = E(say)\\)Now, the net electric field \\(E_P\\) at P due to the dipole is\\(E_P = E_1 + E_2 = 2E\\) ........................(3)Resolving \\(E_1\\) and \\(E_2\\) into horizontal and vertical components, as shown in figure 2. The vertical components will be canceled and horizontal will be added together. So, we can write \\(E_P = 2E \cos \theta \\) .............from(3)\\(E_P = 2 \left (\dfrac{1}{4\pi \epsilon_0} \dfrac{q}{(d^2 + l^2)^2}) (\dfrac{l}{\sqrt (d^2 + l^2)}\right )\\) ............(\\(\because \cos \theta = \dfrac{l}{\sqrt (d^2 + l^2)}\\))For a short dipole, \\(l <<d\\), therefore\\(B_P = \dfrac{1}{4\pi \epsilon_0} \dfrac{2ql}{d^3}\\)\\(B_P = \dfrac{1}{4\pi \epsilon_0} \dfrac{Q}{d^3}\\) ........................(\\(\because Q = 2ql\\))

Two point charges placed at a certain distance r in air exert a force F on each other. The distance r' at which these charges will exert the same force in medium if dielectric constant k is given by

Two point charge placed at a certain distance \\(=r\\)force \\(=F\\)dielectric constant \\(=k\\)Let, two point charges \\({ q }_{ 1 }\\) and \\({ q }_{ 2 }\\) are separated \\(r\\) distance from each other.Then, force experienced between then,\\(F=\dfrac { { q }_{ 1 }{ q }_{ 2 } }{ 4\pi { \epsilon }_{ 0 }{ r }^{ 2 } } \\)Put, \\(F=\dfrac { { q }_{ 1 }{ q }_{ 2 } }{ 4\pi { \epsilon }_{ 0 }{ r }^{ 2 } } \\)\\(\therefore\\) \\(\dfrac { 4{ q }_{ 1 }{ q }_{ 2 } }{ 4\pi { \epsilon }_{ 0 }{ r }^{ 2 } } =\dfrac { { q }_{ 1 }{ q }_{ 2 } }{ k\left( 4\pi { \epsilon }_{ 0 }{ r }^{ 2 } \right) } \\)\\(\Rightarrow \dfrac { 1 }{ { r }^{ 2 } } =\dfrac { 1 }{ 4{ r }^{ 2 }h } \\)\\(\Rightarrow \dfrac { 1 }{ { r }^{ 2 } } =\dfrac { 1 }{ { kr }^{ 2 } } \\)taking square root both sides \\(\dfrac { 1 }{ r } -\dfrac { 1 }{ \sqrt { k } { r }^{ 1 } } \\)or, \\({ r }^{ 1 }=\dfrac { r }{ \sqrt { k } } \\)

Derive an expression for the torque acting on an electric dipole kept in a uniform electric field and hence define electric dipole moment.

Electric dipole moment is the measure of separation of positive and negative charges. It is the measure of overall polarity. Its SI unit is Coulomb-meter.

An electric dipole with dipole moment \\(4\times 10^{-9}\\)C m is aligned at \\(30^\circ\\) with the direction of a uniform electric field of magnitude \\(5\times 10^{4} NC^{-1}\\). Calculate the magnitude of the torque acting on the dipole.

Electric dipole moment, \\( E = 4 \times 10^{-9} N/C\\)Angle made by p with a uniform electric field, θ = 30°Electric field, \\( E = 5 \times 10^4 N/C\\)Torque acting on the dipole: \\(\tau=pE\sin \theta\\)\\(=4\times 10^{-9}\times 5\times 10^4\times sin\, 30^o\\)\\(=20\times 10^{-5}\times \frac{1}{2}\\)\\(=10^{-4}Nm\\)Hence, the magnitude of the torque acting on the dipole is \\(10^{-4}\\) N m.

An electric dipole consist of two opposite charges each of magnitude \\(1\mu C\\) separated by a distance of \\(2\,cm.\\) The dipole is placed in an external field of \\({10^5}{\text{N/C}}\\).The maximum torque on the dipole is:

An electric dipole consist at two opposite charge each of magnitude \\(=1\mu C=1\times { 10 }^{ -6 }C\\)distance \\(=2cm\\)Exter field \\(={ 10 }^{ 5 }N/C\\)maximum torque on the dipole \\(=?\\)\\(q=1\times { 10 }^{ -6 }C,\quad 2a=2cm\\) or, \\(=0.02cm\\)\\(\therefore\\) \\(P=q\times 2a\\) \\(=\left( 1\times { 10 }^{ -6 } \right) \times 0.02\\) \\(=2\times { 10 }^{ -8 }cm\\)Intensity of the external electric field, \\(E=1.0\times { 10 }^{ 5 }N/C\\)(i) \\({ Z }_{ max }=pE=\left( 2\times { 10 }^{ -8 } \right) \left( 10\times { 10 }^{ 5 } \right) =2\times { 10 }^{ -3 }N-m\\)(ii) Net work done in turning the dipole from \\({ 0 }^{ 0 }\\) to \\({ 180 }^{ 0 }\\)i.e \\(W=\int _{ { 0 }^{ 0 } }^{ { 180 }^{ 0 } }{ \overline { r } } d\theta =\int _{ { 0 }^{ 0 } }^{ { 180 }^{ 0 } }{ pE\sin\theta } d\theta \\) \\(=pE{ \left[ -cos\theta \right] }_{ { 0 }^{ 0 } }^{ { 180 }^{ 0 } }\\) \\(=-pE\left( { \cos180 }^{ 0 }-\cos{ 0 }^{ 0 } \right) \\) \\(=2pE\\) \\(=2\times \left( 2\times { 10 }^{ -8 } \right) \left( 1\times { 10 }^{ 5 } \right) J\\) \\(=4\times { 10 }^{ -3 }J\\)

A system has two charges \\(qA = 2.5\times 10^{-7}C\\) and \\(qB = -2.5\times 10^{-7}C\\) located at point \\(A(0, 0, -15) cm\\) and \\(B(0, 0, 15) cm\\) respectively. What are the total charge and electric dipole moment vector of the system?

At A, amount of charge, \\(q_A\\)= \\(2.5\times{10}^{-7}C\\) (Given)At B, amount of charge, \\(q_B= -2.5\times{10}^{-7}\\)Total charge of the system,\\(q=q_A+q_B= 2.5 \times {10}^{-7} C = 0\\)Distance between two charges at points A and B,\\( d=15+15=30 cm = 0.3m\\)Electric dipole moment of the system is given by,\\(p= q_A \times d= 2.5 \times{10}^{-7}\times0.3=7.5\times{10}^{-8}cm\\) along positive z-axis.Therefore, the electric dipole moment of the system is\\( 7.5\times {10}^{-8} \\) along the positive z-axis.

An electric dipole is placed at an angle of \\(30^o\\) with an electric field intensity \\(2\times 10^5\ N/C\\). It experiences a torque equal to \\(4\ Nm.\\) The charge on the dipole, if the dipole length is \\(2\ cm\\), is :

\\(\textbf{Step 1: Dipole moment of dipole}\\)Given the dipole length \\(l = 2\ cm = 2/100\ m\\)So, the Dipole moment of dipole will be \\(P=lq\\) \\( =(2/100)q\\) \\(P=0.02q\\)\\(\textbf{Step 2: Torque acting on dipole }\\) \\(\vec \tau = \vec P \times \vec E \\)\\(\Rightarrow \ \ \tau=PE\sin\theta\\) Or, \\(4\ Nm=(0.02q)(2\times 10^5\ N/C)\sin30^o\\)Or, \\(q=200 \times 10^{-5}C\\) \\( =2 mC\\)Hence, Option C is correct

Electric Dipole | Definition, Examples, Diagrams

Q: Derive an expression for electric field due electric dipole at a point on an equatorial line.

Electric field at A due to the charges \\(q\\) and \\(-q\\) is shown in the above figure. We resolve \\(E\\) into horizontal and vertical components. The vertical components \\((E \sin\theta)\\) cancel out each other and only the horizontal components survive to give net electric field at A as \\(2E\cos\theta\\).Electric field at point A, \\(E_A = 2E \cos\theta\\)where \\(E = \dfrac{q}{4\pi \epsilon_o (r^2 + a^2)}\\)We get \\(E_A = \dfrac{2q }{4\pi \epsilon_o (r^2 + a^2)} \cos\theta\\)From figure, we have \\(\cos\theta = \dfrac{a}{\sqrt{r^2 + a^2}}\\)\\(\therefore\\) \\(E_A = \dfrac{2q a}{4\pi \epsilon_o (r^2 + a^2)^{3/2}} \\)We know dipole moment \\(P = 2qa\\)\\(\implies\\) \\(E_A = \dfrac{P}{4\pi \epsilon_o (r^2 + a^2)^{3/2}} \\)For \\(a<<r\\), we can neglect \\(a^2 \\) compared to \\(r^2\\).\\(\implies\\) \\(E_A = \dfrac{P}{4\pi \epsilon_o r^3} \\)

Q: Derive an expression for the electric field at a point on the equatorial plane of an electric dipole.

Consider an electric dipole AB consisting charges +q and -q separated by distance 2l and pole strength q. Let, P be the point on the equatorial line at distance d from mid-point of the dipole i.e. O as shown in figure 1. Now, the electric field, \\(E_1\\) at point P due to north pole along AP is, \\(E_1 = \dfrac{1}{4\pi \epsilon_0} \dfrac{q}{AP^2}\\) \\(E_1 = \dfrac{1}{4\pi \epsilon_0} \dfrac{q}{(d^2 + l^2)^2}\\) .........(1) (\\(\because AP^2 = OA^2 + OP^2\\)) Similarly, the electric field, \\(E_2\\) at point P due to south pole along PB is, \\(E_2 = \dfrac{1}{4\pi \epsilon_0} \dfrac{q}{PB^2}\\)\\(E_2 = \dfrac{1}{4\pi \epsilon_0} \dfrac{q}{(d^2 + l^2)^2}\\) .........(2) (\\(\because PB^2 = OB^2 + OP^2\\))From equations (1) and (2), we get\\(E_1 = E_2 = E(say)\\)Now, the net electric field \\(E_P\\) at P due to the dipole is\\(E_P = E_1 + E_2 = 2E\\) ........................(3)Resolving \\(E_1\\) and \\(E_2\\) into horizontal and vertical components, as shown in figure 2. The vertical components will be canceled and horizontal will be added together. So, we can write \\(E_P = 2E \cos \theta \\) .............from(3)\\(E_P = 2 \left (\dfrac{1}{4\pi \epsilon_0} \dfrac{q}{(d^2 + l^2)^2}) (\dfrac{l}{\sqrt (d^2 + l^2)}\right )\\) ............(\\(\because \cos \theta = \dfrac{l}{\sqrt (d^2 + l^2)}\\))For a short dipole, \\(l <<d\\), therefore\\(B_P = \dfrac{1}{4\pi \epsilon_0} \dfrac{2ql}{d^3}\\)\\(B_P = \dfrac{1}{4\pi \epsilon_0} \dfrac{Q}{d^3}\\) ........................(\\(\because Q = 2ql\\))

Q: Two point charges placed at a certain distance r in air exert a force F on each other. The distance r' at which these charges will exert the same force in medium if dielectric constant k is given by

Two point charge placed at a certain distance \\(=r\\)force \\(=F\\)dielectric constant \\(=k\\)Let, two point charges \\({ q }_{ 1 }\\) and \\({ q }_{ 2 }\\) are separated \\(r\\) distance from each other.Then, force experienced between then,\\(F=\dfrac { { q }_{ 1 }{ q }_{ 2 } }{ 4\pi { \epsilon }_{ 0 }{ r }^{ 2 } } \\)Put, \\(F=\dfrac { { q }_{ 1 }{ q }_{ 2 } }{ 4\pi { \epsilon }_{ 0 }{ r }^{ 2 } } \\)\\(\therefore\\) \\(\dfrac { 4{ q }_{ 1 }{ q }_{ 2 } }{ 4\pi { \epsilon }_{ 0 }{ r }^{ 2 } } =\dfrac { { q }_{ 1 }{ q }_{ 2 } }{ k\left( 4\pi { \epsilon }_{ 0 }{ r }^{ 2 } \right) } \\)\\(\Rightarrow \dfrac { 1 }{ { r }^{ 2 } } =\dfrac { 1 }{ 4{ r }^{ 2 }h } \\)\\(\Rightarrow \dfrac { 1 }{ { r }^{ 2 } } =\dfrac { 1 }{ { kr }^{ 2 } } \\)taking square root both sides \\(\dfrac { 1 }{ r } -\dfrac { 1 }{ \sqrt { k } { r }^{ 1 } } \\)or, \\({ r }^{ 1 }=\dfrac { r }{ \sqrt { k } } \\)

Q: Derive an expression for the torque acting on an electric dipole kept in a uniform electric field and hence define electric dipole moment.

Electric dipole moment is the measure of separation of positive and negative charges. It is the measure of overall polarity. Its SI unit is Coulomb-meter.

Q: An electric dipole with dipole moment \\(4\times 10^{-9}\\)C m is aligned at \\(30^\circ\\) with the direction of a uniform electric field of magnitude \\(5\times 10^{4} NC^{-1}\\). Calculate the magnitude of the torque acting on the dipole.

Electric dipole moment, \\( E = 4 \times 10^{-9} N/C\\)Angle made by p with a uniform electric field, θ = 30°Electric field, \\( E = 5 \times 10^4 N/C\\)Torque acting on the dipole: \\(\tau=pE\sin \theta\\)\\(=4\times 10^{-9}\times 5\times 10^4\times sin\, 30^o\\)\\(=20\times 10^{-5}\times \frac{1}{2}\\)\\(=10^{-4}Nm\\)Hence, the magnitude of the torque acting on the dipole is \\(10^{-4}\\) N m.

Q: An electric dipole consist of two opposite charges each of magnitude \\(1\mu C\\) separated by a distance of \\(2\,cm.\\) The dipole is placed in an external field of \\({10^5}{\text{N/C}}\\).The maximum torque on the dipole is:

An electric dipole consist at two opposite charge each of magnitude \\(=1\mu C=1\times { 10 }^{ -6 }C\\)distance \\(=2cm\\)Exter field \\(={ 10 }^{ 5 }N/C\\)maximum torque on the dipole \\(=?\\)\\(q=1\times { 10 }^{ -6 }C,\quad 2a=2cm\\) or, \\(=0.02cm\\)\\(\therefore\\) \\(P=q\times 2a\\) \\(=\left( 1\times { 10 }^{ -6 } \right) \times 0.02\\) \\(=2\times { 10 }^{ -8 }cm\\)Intensity of the external electric field, \\(E=1.0\times { 10 }^{ 5 }N/C\\)(i) \\({ Z }_{ max }=pE=\left( 2\times { 10 }^{ -8 } \right) \left( 10\times { 10 }^{ 5 } \right) =2\times { 10 }^{ -3 }N-m\\)(ii) Net work done in turning the dipole from \\({ 0 }^{ 0 }\\) to \\({ 180 }^{ 0 }\\)i.e \\(W=\int _{ { 0 }^{ 0 } }^{ { 180 }^{ 0 } }{ \overline { r } } d\theta =\int _{ { 0 }^{ 0 } }^{ { 180 }^{ 0 } }{ pE\sin\theta } d\theta \\) \\(=pE{ \left[ -cos\theta \right] }_{ { 0 }^{ 0 } }^{ { 180 }^{ 0 } }\\) \\(=-pE\left( { \cos180 }^{ 0 }-\cos{ 0 }^{ 0 } \right) \\) \\(=2pE\\) \\(=2\times \left( 2\times { 10 }^{ -8 } \right) \left( 1\times { 10 }^{ 5 } \right) J\\) \\(=4\times { 10 }^{ -3 }J\\)

Q: A system has two charges \\(qA = 2.5\times 10^{-7}C\\) and \\(qB = -2.5\times 10^{-7}C\\) located at point \\(A(0, 0, -15) cm\\) and \\(B(0, 0, 15) cm\\) respectively. What are the total charge and electric dipole moment vector of the system?

At A, amount of charge, \\(q_A\\)= \\(2.5\times{10}^{-7}C\\) (Given)At B, amount of charge, \\(q_B= -2.5\times{10}^{-7}\\)Total charge of the system,\\(q=q_A+q_B= 2.5 \times {10}^{-7} C = 0\\)Distance between two charges at points A and B,\\( d=15+15=30 cm = 0.3m\\)Electric dipole moment of the system is given by,\\(p= q_A \times d= 2.5 \times{10}^{-7}\times0.3=7.5\times{10}^{-8}cm\\) along positive z-axis.Therefore, the electric dipole moment of the system is\\( 7.5\times {10}^{-8} \\) along the positive z-axis.

Q: An electric dipole is placed at an angle of \\(30^o\\) with an electric field intensity \\(2\times 10^5\ N/C\\). It experiences a torque equal to \\(4\ Nm.\\) The charge on the dipole, if the dipole length is \\(2\ cm\\), is :

\\(\textbf{Step 1: Dipole moment of dipole}\\)Given the dipole length \\(l = 2\ cm = 2/100\ m\\)So, the Dipole moment of dipole will be \\(P=lq\\) \\( =(2/100)q\\) \\(P=0.02q\\)\\(\textbf{Step 2: Torque acting on dipole }\\) \\(\vec \tau = \vec P \times \vec E \\)\\(\Rightarrow \ \ \tau=PE\sin\theta\\) Or, \\(4\ Nm=(0.02q)(2\times 10^5\ N/C)\sin30^o\\)Or, \\(q=200 \times 10^{-5}C\\) \\( =2 mC\\)Hence, Option C is correct

definition

Physical significance of dipoles

In most molecules, the centres of positive charges and of negative charges lie at the same place. Therefore, their dipole moment is zero.

C O_{2}

and

C H_{4}

are of this type of molecules. However, they develop a dipole moment when an electric field is applied. But in some molecules, the centres of negative charges and of positive charges do not coincide. Therefore they have a permanent electric dipole moment, even in the absence of an electric field. Such molecules are called polar molecules. Water molecules,

H_{2} O

,is an example of this type. Various materials give rise to interesting properties and important applications in the presence or absence of electric field.

definition

Electric dipole

An electric dipole is a pair of equal and opposite point charges -q and q, separated by a distance 2a. The direction from q to -q is said to be the direction of the dipole.

p = q \times 2 a

where

p

is the electric dipole moment pointing from the negative charge to the positive charge.

example

Force on electric dipole

A small electric dipole having dipole moment

p

is placed along X-axis, as shown in the figure. A semi-infinite uniformly charged di-electric thin rod placed along x axis, with one end coinciding with origin. If linear charge density rod is

+ λ

and distance of dipole from rod is

^{'} a^{'}

, then calculate the electric force acting on dipole.
Electric field at A due to dipole,

E_{A} = \frac{2 K P}{r ^{3}}

where

K = \frac{1}{4 π ϵ _{o}}

E_{A} = \frac{2 K P}{( x + a ) ^{3}}

Force at A due to dipole,

F_{A} = q E_{A} = (λ

d x)

\times \frac{2 K P}{( x + a ) ^{3}}

F_{A} = \frac{2 λ K P}{( x + a ) ^{3}} d x

Total force exerted due to dipole on the rod,

F_{D} = \int_{0}^{\infty} \frac{2 λ K P}{( x + a ) ^{3}} d x

F_{D} = 2 λ K P \times \frac{1}{- 2 ( x + a ) ^{2}} ∣ ∣ ∣ ∣ ∣_{0}^{\infty}

⟹ F_{D} = \frac{λ K P}{a ^{2}}

Now force exerted by rod on dipole is equal and opposite to

F_{D}

(Newton's law)

F_{R} = - F_{D} = \frac{- λ P}{4 π ϵ _{o} a ^{2}}

Electric Dipole

definition

Physical significance of dipoles

definition

Electric dipole

example

Force on electric dipole

REVISE WITH CONCEPTS

Field Due to Electric Dipole

Potential Due to an Electric Dipole

Potential Energy of a Dipole in External Field

Torque on a Dipole Placed in an Electric Field

Quick Summary With Stories

Electric Dipole and Dipole Moment

Derive an expression for electric field due to an electric dipole at a point on its axial line.

Derive an expression for electric potential at a point due to an electric dipole. Discuss the special cases.

Derive an expression for electric field due electric dipole at a point on an equatorial line.

Derive an expression for the electric field at a point on the equatorial plane of an electric dipole.

Two point charges placed at a certain distance r in air exert a force F on each other. The distance r' at which these charges will exert the same force in medium if dielectric constant k is given by

Derive an expression for the torque acting on an electric dipole kept in a uniform electric field and hence define electric dipole moment.

An electric dipole with dipole moment $4 \times 1 0^{- 9}$ C m is aligned at $3 0^{\circ}$ with the direction of a uniform electric field of magnitude $5 \times 1 0^{4} N C^{- 1}$ . Calculate the magnitude of the torque acting on the dipole.

An electric dipole consist of two opposite charges each of magnitude $1 μ C$ separated by a distance of $2 c m .$ The dipole is placed in an external field of $1 0^{5} N/C$ .The maximum torque on the dipole is:

A system has two charges $q A = 2.5 \times 1 0^{- 7} C$ and $q B = - 2.5 \times 1 0^{- 7} C$ located at point $A (0, 0, - 15) c m$ and $B (0, 0, 15) c m$ respectively. What are the total charge and electric dipole moment vector of the system?

An electric dipole is placed at an angle of $3 0^{o}$ with an electric field intensity $2 \times 1 0^{5} N / C$ . It experiences a torque equal to $4 N m .$ The charge on the dipole, if the dipole length is $2 c m$ , is :

definition

Physical significance of dipoles

definition

Electric dipole

example

Force on electric dipole

REVISE WITH CONCEPTS

Field Due to Electric Dipole

Potential Due to an Electric Dipole

Potential Energy of a Dipole in External Field

Torque on a Dipole Placed in an Electric Field

Quick Summary With Stories

Electric Dipole and Dipole Moment

Derive an expression for electric field due to an electric dipole at a point on its axial line.

Derive an expression for electric potential at a point due to an electric dipole. Discuss the special cases.

Derive an expression for electric field due electric dipole at a point on an equatorial line.

Derive an expression for the electric field at a point on the equatorial plane of an electric dipole.

Two point charges placed at a certain distance r in air exert a force F on each other. The distance r' at which these charges will exert the same force in medium if dielectric constant k is given by

Derive an expression for the torque acting on an electric dipole kept in a uniform electric field and hence define electric dipole moment.

An electric dipole with dipole moment 4×10−9C m is aligned at 30∘ with the direction of a uniform electric field of magnitude 5×104NC−1. Calculate the magnitude of the torque acting on the dipole.

An electric dipole consist of two opposite charges each of magnitude 1μC separated by a distance of 2cm. The dipole is placed in an external field of 105N/C.The maximum torque on the dipole is:

A system has two charges qA=2.5×10−7C and qB=−2.5×10−7C located at point A(0,0,−15)cm and B(0,0,15)cm respectively. What are the total charge and electric dipole moment vector of the system?

An electric dipole is placed at an angle of 30o with an electric field intensity 2×105 N/C. It experiences a torque equal to 4 Nm. The charge on the dipole, if the dipole length is 2 cm, is :

An electric dipole with dipole moment $4 \times 1 0^{- 9}$ C m is aligned at $3 0^{\circ}$ with the direction of a uniform electric field of magnitude $5 \times 1 0^{4} N C^{- 1}$ . Calculate the magnitude of the torque acting on the dipole.

An electric dipole consist of two opposite charges each of magnitude $1 μ C$ separated by a distance of $2 c m .$ The dipole is placed in an external field of $1 0^{5} N/C$ .The maximum torque on the dipole is:

A system has two charges $q A = 2.5 \times 1 0^{- 7} C$ and $q B = - 2.5 \times 1 0^{- 7} C$ located at point $A (0, 0, - 15) c m$ and $B (0, 0, 15) c m$ respectively. What are the total charge and electric dipole moment vector of the system?

An electric dipole is placed at an angle of $3 0^{o}$ with an electric field intensity $2 \times 1 0^{5} N / C$ . It experiences a torque equal to $4 N m .$ The charge on the dipole, if the dipole length is $2 c m$ , is :