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CURRENT ELECTRICITY

2.1

CURRENT ELECTRICITY

1

Electric Current Rate of folow of electric charge, SI Unit is ampare (A). I=

q t

, where 'q' is the total charge and 't' is the time period. 1coulomb

1 ampere = 1sec ond

1 ampare is defined as the current in a wire if one coulomb of charges flows through it in one second. 2.2

Drift Velocity In conductors, a large number of free electrons are present, which move in random directions. If r r r r u1,u2 ,u3 ,.....uN

are their random initial velocities, then the average thermal velocity is

r r r r u1,u2 ,u3 ,.....uN N

=0

.........(1) A +

E

l

F = –eE

_

+ _

r

When an external electric field E is applied across the conductor, the electrons experience a force, r r F = –eE

.........(2)

If me be the mass of electron, then the acceleration of electron will be, r r - eE a= me

.........(3)

Under the in infuluence of electri field, electron move and subsequently collide with the tions. The time between two successive collisiuons is called Relaxation time ( t ). he velocity of elecytron just before collision is given by, r v1

r

r

= u1 + a t1

r

.........(4)

r

Where v1 & u1 are the final and initial velocities and t1is the relaxation time of electron I. The drift velocity is defined as the average velocity with which free electrons in a conductor get drifted r

under the influence of an external field. It is represented by v d . r r r r v1 + v 2 + v 3 + ..... + v N N r r r r u1 + u2 + u3 + ......... + uN r t1 + t2 + ..........tN +a N N r

Hence, v d =

t1 + t2 + ..........tN N r r v d = at

Where

\

r r -eE vd = t me

.........(5)

= t, the average relaxation time. Using equation (1), we get,

........(6)

This represents the equation for drift velocity. The drift velocity for metals is of the order of 10–3 m/s » 10–5 m/s

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2 2.3

Relation between Drift Velocity and Electric Current

CURRENT ELECTRICITY

Let us consider the conductor of cross section area A and length l. Then the volume of the conductor, V =Al If n be the no. of free electrons per unit volme, then the total charge on all the free electrons is given by, q = nAl e

.........(1)

Under the influence of electric field, the electrons will travel the distance l with drift velocity vd in

2.4

l q t = v ÛI = t d

.........(2)

I = neA vd

.........(3)

Mobility The mobility of free electrons is defined as the drift velocity acquired per unit strength of applied electric field. It is represented by m. Thus, m=

\

vd E

I = nA m E e

SI unit of mobility m2V–1s–1 3.

OHM'S LAW The current through a conductor is always directly proportional to the p. d. across its two ends, subject to particular physical conditions (temperature, pressure, etc.) Mathematically. I=V Or we can say that,

V=I V = RI , where R is the resistance.

Thus,

R=

V I

The SI unit of resistance is W (ohm). 1 ohm: On applying a p. d. of 1 volt across the conductor, if one ampare current flows through it, the resistance of a wire is said to be 1 ohm. Failure of Ohm's Law 1.

Due to joule's Effect, the increase in current causes an increase in temperature in the conductor. Hence3 the resistance in the wire in the wire increases. Thus, current does not actually vary linearly with potential difference. Fig. 1.

V(Volts)

Ohm's Law being followed Failure of Ohm's Law

I(Amp)

Fig.1

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CURRENT ELECTRICITY 3 2. When a semiconductor diode is applied with a p. d. across it, the amount of current crossing the diode depends upon the direction in which current flows. In forward biased condition, the current that flows, is given by Fig. 2 whereas the current that flows during reverse biasing is shown in Fig. 3. 3.

In electrolytes, the Ohm's Law fails.

4.

In semiconductor devices, such as thyristors, the current may increase on decreasing the p. d. Fig.4. V(Volts) I (mA)

I (mA)

–I (mA)

V(Volts)

V(Volts)

Fig.2

Fig.3

Fig.4 Non linear Negative positive Resistance Resistance Region Region

I (mA)

V(Volts) Variation of current versus valtage for GaAs.

3.1

Resistivity (Specific Resistance) The resistance of a conductor depends upon factors such as length and cross section area of the conductor. (a) It is directly proportional to the length of the conductor, i.e., R µl (b) It is inversely proportional to the area of cross section of the conductor Rµ

1 , Hence, A

R=r

l A

where, r is the resistivity of the conductor.

Metal of least : 1. Silver 1.6×10–8 m 2. Copper 1.7×10–8 m Alloy Metal of high : 1. Nichrome 100×10–8 m 2. Mercury 98×10–8 m

Resistivity is defined as the resistance of a material of a conductor of unit length and unit area of cross section. SI unit of resistivity is W m. Now, V = IR= I r Þ \

V=

I A

l A

rl = jrl [where j ® current density] Material with highest resistivit y

E = jr Þ j = sE

Fused quartz 1016 Wm

where s is the conductivity and is reciprocal of r. 3.2

Material which is sup erconducti ng Tl2 Ba2 Ca2 Cu2 O10

Resistivity

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4

CURRENT ELECTRICITY The drift velocity of electron V l

Þ vd =

Þ

E=

Þ

I = n A e vd

\

I=

R=

V I

t

I = nevd A E j = = sE ........( 2) r

j=

From equations (1) and ( 2 )

r =

j=

m

l A ne t 2

As we know, R = r

ne 2 t E m

and s =

ne 2 t m

l A

m ne2 t

Temperature Dependence of Resistivity: The relaxation time t decreases with the increase in temperature. Thus, an increase in the temperature will result in higher resistivity of the material of the conductor. There are two formual, which explain this relation: 1 r µ ; r = r0 [1 + a(T - T0 )] t

3.4

eV ml

j ® Current density

nAe 2 Vt ml

But, R =

3.3

r r eE vd = t m

T Û t ¯Û r Û R

Dependence on Material : The no. of free electrons per unit volume is 'n' and it varies from material. The resistivity of the conductor also depends upon the material. rµ

3.5

1 n

Conductance and Conductivity: Conductance is defined as the reciprocal of resistance of the material. It is represented by G. The SI Unit of conductance is Ohm–1 [or Mho or W –1 or Siemens (S)] G=

1 R

Wh y ma ng a ni n i s u s e d i n t h e construction of standard resistance coil? Alloys such as manganin have very high resistivity and weak dependence on themperature. Manganin (An alloy of copper, Nickel, Iron and Manganese has 30 to 40 times greater resistivity and temperature coefficient is 0.00002ºC–1. Therefore, the resistance of the coil does not change appreciably due to Joule's Heating Effect.

3.6

Conductivity: Conductivity is defined as the reciprocal of resitivity of the meterial. It is represented by s.

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CURRENT ELECTRICITY The SI unit of conductance is Ohm–1m–1 (or W–1 m–1 or Siemen m–1) s=

3.7

5

1 r

Temerature Coefficient of Resistance Let r and r0 be the resistivity of the conducting matrial at qºC and 0º C respectively. Then, r = r0 (1 + aq)

Material like Nichrome exhibit a very weak dependence of resistivity Manganin and constantan have similar properties l Since,R = r , we can also write, A

R = R0( 1 + 1aq) \ 3.8

a=

R - R0 R 0q

For metals a lies between 10 -2 to 10 -4 º C -1

Conduction in semiconductors : It is due to both electrons and holes. Hence, s = neme + pemh ; me and mh ® mobility of electron and hole n and p ® electron and hole concentration. nd =

eE t m

me =

ete me

and m =

nd E

et h and m h = m h

Temperature Vs Resistivity For conductors, at low temperature, resistivity increases as a higher power of temperature. Resistivity, r µ r0 [1 + a(T - T0 )].

T

Resistivity Vs temperature(Cu)

T

Resistivity Vs temperature (Nichrome)

T

Conductivity Vs Temp (Conductor)

Temperature Vs Resistivity (for Semiconductors and Insulators) In case of semiconductor and an insulator, the resistivity at an absolute temperature T, is given by, Eg

r = r 0 e kT

where, Eg ® Band Gap Energy; and k ® Bolzmann Constant Superconductivity: The phenomenon, due to which a substance loses its resistance, when cooled below its critical temperature, is known as superconductivity. The substance, which allows current to apss through it without any resistance, is known as superconductor. When Hg is cooled to 4.2 K or below, it loses all its property of resistance.

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T

Resistivity Vs Temp. (Semiconductor) (or graphite)

6

CURRENT ELECTRICITY In 1988, superconductivity was achieved with superconducting oxides at 125K. The cause of superconductivity is that electrons in superconductor are mutually coherent. The ionic vibrations, which can easily. deflect free electrons in metals, are unable to deflect the coherent cloud of electrons in superconductors. Applications 1. It can be used to produce super fast computers. 4.2 K T Resistivity Vs Temperature(Hg) 2. It can be used to avoid power loss during transmission 3.It can be used to produce very high magnetic fields without spending electrical power.

Pneum onic B B R O Y Gets Beautiful & Very Good W ife Gold Silver

Colour Black Brown Red Orange Yellow Green Blue Violet Grey W hite Gold Silver

Value 0 1 2 3 4 5 6 7 8 9 10 –1 10 –2

Colour Gold Silver No Colour

Tolerance ± 5% ± 10% ± 20%

Colour Code for Carbon Resistors A BC

D

Carbon Resistor

The Formula, C AB × 10 ± D%

If A = Red, B = Violet, C= Yellow and D = Gold, then the resistance of the carbon resistor is 27 ×104 W ± 5% D represents 'tolerance', which means possible variation in % about the specified value 4.

Resistors in Series Consider three resistors R1, R2 & R3 in series, A battary of EMF E is connected across the combination such that a current 'I' passes through the circuit giving rise to p. d. of V 1, v2 & V3 across R1, R2 & R3 respectively. Hence R1

R2

R3

V1

V2

V3

V I

+

_ E

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CURRENT ELECTRICITY

7

V1 = IR1; V2 = IR2 ; V3 = IR3 V = V1 + V2 + V3 IReq = IR1 + IR2 + IR3 Req = R1 + R2 + R3

5.

Resistors in Parallel Consider three resistors R1, R2 & R3 put in parallel combination. A battery of EMF E is connected across the combination. Then the current ' I ' in the circuit can be given by I = I1 + I2 + I3

R1 I1

R2

I2

I3

R3

whre I1, I2 and I3 represent the current through R1, R2 & R3 respectively. But, the p. d. across each resistor is V so, I=

V V V V = + + R eq R 1 R 2 R 3

V I

+

1 1 1 1 = + + R eq R1 R 2 R 3

Internal Resistance of Cell: The resistance offered by the electrolyte of the cell to the flow of current. It is repesented by r. It depends upon (a) Nature of electrolyte (b) Distance between electrodes (c) Area of electrode (d) Concentration of electrolyte EMF(Electromotive Force) : The p. d. across the terminals of a cell when no current u being drawn by the circuit. It is denoted by E. SI unit is volt. Relation between EMF (E) & (V) The EMF is given by , E = I(R + r) \

I=

E R+r

where r is the internal resistance of the cell. Þ

æ E ö V =ç ÷R èR+rø

Þ Þ

V(R + r) = ER Vr = (E – V)R

Þ

æE-Vö r =ç ÷R è V ø

1 Volt: The EMF of a cell is one Volt, if one Joule of work is performed by the cell to drive one Coulomb of charge round the circuit. Grouping of Cells Cells in Series: Consider n cells of EMF E and internal resistance r.

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8

CURRENT ELECTRICITY E I

E

r

E

r

I

r

For different cells in series, E1 + E 2 + E 3+......... I= (r1 + r2 + r3 + .......)+ R For different cells in parallel, E eq E1 E 2 = + + .......... . req r1 r2

R Cells in Series

Then the total EMF = nE The total resistance = nr + R nE Hence, current in the circuit, I =

R + nr

If out of n cells, m cells are grouped in reverse order then, net EMF = nE–(2m)E Cells in Parallel : Consider m rows of cells of EMF E and internal resistance r. Then the EMF = E The total resistance =

Hence, current in the circuit, I =

I=

mE mR + r

If r

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2.1

CURRENT ELECTRICITY

1

Electric Current Rate of folow of electric charge, SI Unit is ampare (A). I=

q t

, where 'q' is the total charge and 't' is the time period. 1coulomb

1 ampere = 1sec ond

1 ampare is defined as the current in a wire if one coulomb of charges flows through it in one second. 2.2

Drift Velocity In conductors, a large number of free electrons are present, which move in random directions. If r r r r u1,u2 ,u3 ,.....uN

are their random initial velocities, then the average thermal velocity is

r r r r u1,u2 ,u3 ,.....uN N

=0

.........(1) A +

E

l

F = –eE

_

+ _

r

When an external electric field E is applied across the conductor, the electrons experience a force, r r F = –eE

.........(2)

If me be the mass of electron, then the acceleration of electron will be, r r - eE a= me

.........(3)

Under the in infuluence of electri field, electron move and subsequently collide with the tions. The time between two successive collisiuons is called Relaxation time ( t ). he velocity of elecytron just before collision is given by, r v1

r

r

= u1 + a t1

r

.........(4)

r

Where v1 & u1 are the final and initial velocities and t1is the relaxation time of electron I. The drift velocity is defined as the average velocity with which free electrons in a conductor get drifted r

under the influence of an external field. It is represented by v d . r r r r v1 + v 2 + v 3 + ..... + v N N r r r r u1 + u2 + u3 + ......... + uN r t1 + t2 + ..........tN +a N N r

Hence, v d =

t1 + t2 + ..........tN N r r v d = at

Where

\

r r -eE vd = t me

.........(5)

= t, the average relaxation time. Using equation (1), we get,

........(6)

This represents the equation for drift velocity. The drift velocity for metals is of the order of 10–3 m/s » 10–5 m/s

www.thinkiit.in

2 2.3

Relation between Drift Velocity and Electric Current

CURRENT ELECTRICITY

Let us consider the conductor of cross section area A and length l. Then the volume of the conductor, V =Al If n be the no. of free electrons per unit volme, then the total charge on all the free electrons is given by, q = nAl e

.........(1)

Under the influence of electric field, the electrons will travel the distance l with drift velocity vd in

2.4

l q t = v ÛI = t d

.........(2)

I = neA vd

.........(3)

Mobility The mobility of free electrons is defined as the drift velocity acquired per unit strength of applied electric field. It is represented by m. Thus, m=

\

vd E

I = nA m E e

SI unit of mobility m2V–1s–1 3.

OHM'S LAW The current through a conductor is always directly proportional to the p. d. across its two ends, subject to particular physical conditions (temperature, pressure, etc.) Mathematically. I=V Or we can say that,

V=I V = RI , where R is the resistance.

Thus,

R=

V I

The SI unit of resistance is W (ohm). 1 ohm: On applying a p. d. of 1 volt across the conductor, if one ampare current flows through it, the resistance of a wire is said to be 1 ohm. Failure of Ohm's Law 1.

Due to joule's Effect, the increase in current causes an increase in temperature in the conductor. Hence3 the resistance in the wire in the wire increases. Thus, current does not actually vary linearly with potential difference. Fig. 1.

V(Volts)

Ohm's Law being followed Failure of Ohm's Law

I(Amp)

Fig.1

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CURRENT ELECTRICITY 3 2. When a semiconductor diode is applied with a p. d. across it, the amount of current crossing the diode depends upon the direction in which current flows. In forward biased condition, the current that flows, is given by Fig. 2 whereas the current that flows during reverse biasing is shown in Fig. 3. 3.

In electrolytes, the Ohm's Law fails.

4.

In semiconductor devices, such as thyristors, the current may increase on decreasing the p. d. Fig.4. V(Volts) I (mA)

I (mA)

–I (mA)

V(Volts)

V(Volts)

Fig.2

Fig.3

Fig.4 Non linear Negative positive Resistance Resistance Region Region

I (mA)

V(Volts) Variation of current versus valtage for GaAs.

3.1

Resistivity (Specific Resistance) The resistance of a conductor depends upon factors such as length and cross section area of the conductor. (a) It is directly proportional to the length of the conductor, i.e., R µl (b) It is inversely proportional to the area of cross section of the conductor Rµ

1 , Hence, A

R=r

l A

where, r is the resistivity of the conductor.

Metal of least : 1. Silver 1.6×10–8 m 2. Copper 1.7×10–8 m Alloy Metal of high : 1. Nichrome 100×10–8 m 2. Mercury 98×10–8 m

Resistivity is defined as the resistance of a material of a conductor of unit length and unit area of cross section. SI unit of resistivity is W m. Now, V = IR= I r Þ \

V=

I A

l A

rl = jrl [where j ® current density] Material with highest resistivit y

E = jr Þ j = sE

Fused quartz 1016 Wm

where s is the conductivity and is reciprocal of r. 3.2

Material which is sup erconducti ng Tl2 Ba2 Ca2 Cu2 O10

Resistivity

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4

CURRENT ELECTRICITY The drift velocity of electron V l

Þ vd =

Þ

E=

Þ

I = n A e vd

\

I=

R=

V I

t

I = nevd A E j = = sE ........( 2) r

j=

From equations (1) and ( 2 )

r =

j=

m

l A ne t 2

As we know, R = r

ne 2 t E m

and s =

ne 2 t m

l A

m ne2 t

Temperature Dependence of Resistivity: The relaxation time t decreases with the increase in temperature. Thus, an increase in the temperature will result in higher resistivity of the material of the conductor. There are two formual, which explain this relation: 1 r µ ; r = r0 [1 + a(T - T0 )] t

3.4

eV ml

j ® Current density

nAe 2 Vt ml

But, R =

3.3

r r eE vd = t m

T Û t ¯Û r Û R

Dependence on Material : The no. of free electrons per unit volume is 'n' and it varies from material. The resistivity of the conductor also depends upon the material. rµ

3.5

1 n

Conductance and Conductivity: Conductance is defined as the reciprocal of resistance of the material. It is represented by G. The SI Unit of conductance is Ohm–1 [or Mho or W –1 or Siemens (S)] G=

1 R

Wh y ma ng a ni n i s u s e d i n t h e construction of standard resistance coil? Alloys such as manganin have very high resistivity and weak dependence on themperature. Manganin (An alloy of copper, Nickel, Iron and Manganese has 30 to 40 times greater resistivity and temperature coefficient is 0.00002ºC–1. Therefore, the resistance of the coil does not change appreciably due to Joule's Heating Effect.

3.6

Conductivity: Conductivity is defined as the reciprocal of resitivity of the meterial. It is represented by s.

www.thinkiit.in

CURRENT ELECTRICITY The SI unit of conductance is Ohm–1m–1 (or W–1 m–1 or Siemen m–1) s=

3.7

5

1 r

Temerature Coefficient of Resistance Let r and r0 be the resistivity of the conducting matrial at qºC and 0º C respectively. Then, r = r0 (1 + aq)

Material like Nichrome exhibit a very weak dependence of resistivity Manganin and constantan have similar properties l Since,R = r , we can also write, A

R = R0( 1 + 1aq) \ 3.8

a=

R - R0 R 0q

For metals a lies between 10 -2 to 10 -4 º C -1

Conduction in semiconductors : It is due to both electrons and holes. Hence, s = neme + pemh ; me and mh ® mobility of electron and hole n and p ® electron and hole concentration. nd =

eE t m

me =

ete me

and m =

nd E

et h and m h = m h

Temperature Vs Resistivity For conductors, at low temperature, resistivity increases as a higher power of temperature. Resistivity, r µ r0 [1 + a(T - T0 )].

T

Resistivity Vs temperature(Cu)

T

Resistivity Vs temperature (Nichrome)

T

Conductivity Vs Temp (Conductor)

Temperature Vs Resistivity (for Semiconductors and Insulators) In case of semiconductor and an insulator, the resistivity at an absolute temperature T, is given by, Eg

r = r 0 e kT

where, Eg ® Band Gap Energy; and k ® Bolzmann Constant Superconductivity: The phenomenon, due to which a substance loses its resistance, when cooled below its critical temperature, is known as superconductivity. The substance, which allows current to apss through it without any resistance, is known as superconductor. When Hg is cooled to 4.2 K or below, it loses all its property of resistance.

www.thinkiit.in

T

Resistivity Vs Temp. (Semiconductor) (or graphite)

6

CURRENT ELECTRICITY In 1988, superconductivity was achieved with superconducting oxides at 125K. The cause of superconductivity is that electrons in superconductor are mutually coherent. The ionic vibrations, which can easily. deflect free electrons in metals, are unable to deflect the coherent cloud of electrons in superconductors. Applications 1. It can be used to produce super fast computers. 4.2 K T Resistivity Vs Temperature(Hg) 2. It can be used to avoid power loss during transmission 3.It can be used to produce very high magnetic fields without spending electrical power.

Pneum onic B B R O Y Gets Beautiful & Very Good W ife Gold Silver

Colour Black Brown Red Orange Yellow Green Blue Violet Grey W hite Gold Silver

Value 0 1 2 3 4 5 6 7 8 9 10 –1 10 –2

Colour Gold Silver No Colour

Tolerance ± 5% ± 10% ± 20%

Colour Code for Carbon Resistors A BC

D

Carbon Resistor

The Formula, C AB × 10 ± D%

If A = Red, B = Violet, C= Yellow and D = Gold, then the resistance of the carbon resistor is 27 ×104 W ± 5% D represents 'tolerance', which means possible variation in % about the specified value 4.

Resistors in Series Consider three resistors R1, R2 & R3 in series, A battary of EMF E is connected across the combination such that a current 'I' passes through the circuit giving rise to p. d. of V 1, v2 & V3 across R1, R2 & R3 respectively. Hence R1

R2

R3

V1

V2

V3

V I

+

_ E

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CURRENT ELECTRICITY

7

V1 = IR1; V2 = IR2 ; V3 = IR3 V = V1 + V2 + V3 IReq = IR1 + IR2 + IR3 Req = R1 + R2 + R3

5.

Resistors in Parallel Consider three resistors R1, R2 & R3 put in parallel combination. A battery of EMF E is connected across the combination. Then the current ' I ' in the circuit can be given by I = I1 + I2 + I3

R1 I1

R2

I2

I3

R3

whre I1, I2 and I3 represent the current through R1, R2 & R3 respectively. But, the p. d. across each resistor is V so, I=

V V V V = + + R eq R 1 R 2 R 3

V I

+

1 1 1 1 = + + R eq R1 R 2 R 3

Internal Resistance of Cell: The resistance offered by the electrolyte of the cell to the flow of current. It is repesented by r. It depends upon (a) Nature of electrolyte (b) Distance between electrodes (c) Area of electrode (d) Concentration of electrolyte EMF(Electromotive Force) : The p. d. across the terminals of a cell when no current u being drawn by the circuit. It is denoted by E. SI unit is volt. Relation between EMF (E) & (V) The EMF is given by , E = I(R + r) \

I=

E R+r

where r is the internal resistance of the cell. Þ

æ E ö V =ç ÷R èR+rø

Þ Þ

V(R + r) = ER Vr = (E – V)R

Þ

æE-Vö r =ç ÷R è V ø

1 Volt: The EMF of a cell is one Volt, if one Joule of work is performed by the cell to drive one Coulomb of charge round the circuit. Grouping of Cells Cells in Series: Consider n cells of EMF E and internal resistance r.

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8

CURRENT ELECTRICITY E I

E

r

E

r

I

r

For different cells in series, E1 + E 2 + E 3+......... I= (r1 + r2 + r3 + .......)+ R For different cells in parallel, E eq E1 E 2 = + + .......... . req r1 r2

R Cells in Series

Then the total EMF = nE The total resistance = nr + R nE Hence, current in the circuit, I =

R + nr

If out of n cells, m cells are grouped in reverse order then, net EMF = nE–(2m)E Cells in Parallel : Consider m rows of cells of EMF E and internal resistance r. Then the EMF = E The total resistance =

Hence, current in the circuit, I =

I=

mE mR + r

If r

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