A parallel-plate capacitor with area of each plate equal to S , and the separation between them, to d , is put into a stream of conducting liquid with resistivity ρ . The liquid moves parallel to the plates with a constant velocity v → . The whole system is located in a uniform magnetic field of induction B → , vector B → being parallel to the plates and perpendicular to the stream direction. The capacitor plates are interconnected by means of an external resistance R . What amount of power is generated in that resistance? At what value of R , is the generated power the highest? What is this highest power equal to?

As the conducting liquid consists of positive and negative ions and the system is placed here in an external magnetic field. Therefore, there will be motional emf generated between the plate of capacitor. The resistance of the liquid between the plates = ρ d S The voltage between the plates = E d = v B d The current through the plates = v B d R + ρ d S The power generated in the external resistance is P = v 2 B 2 d 2 R R + ρ d S 2 = v 2 B 2 d 2 R + ρ d S R 2 = v 2 B 2 d 2 R 1 4 - ρ d S R 1 2 2 + 2 ρ d S 2 For maximum power generated, d P d R = 0 . This is maximum when R = ρ D S and P max = v 2 B 2 S d 4 ρ .

A small current-carrying loop is located at a distance r from a long straight conductor with current I . The magnetic moment of the loop is equal to p m → . Find the magnitude and direction of the force vector applied to the loop, if p m → (a) is parallel to the straight conductor, (b) is oriented along the radius vector r → , (c) coincides in the direction with the magnetic field produced by the current I , at the point where the loop is located.

The magnetic field due to the straight conductor, at a perpendicular distance r is B = μ 0 I 2 π r , where, B = magnetic field, I = current, r = the radius of the loop, μ 0 = the permeability of free space. The force acting on the loop is, F → = p m → · ∇ → B → , where, p m → = the magnetic moment of the loop, ∇ → = Differential = ∂ ∂ x i ^ + ∂ ∂ y j ^ + ∂ ∂ z k ^ . (a) Now, given that the vector p m → is parallel to the straight conductor, F → = p m ∂ ∂ Z B → = 0 . This is because neither the direction nor the magnitude of B → depends on z . (b) The vector p m → is oriented along the radius vector r → . So, for this, ∇ for the cylindrical co-ordinates can be given as, ∇ → = ∂ ∂ r r ^ + 1 r ∂ ∂ φ φ ^ + ∂ ∂ z k ^ , F → = p m ∂ ∂ r B → . The direction of B → at r + d r , is parallel to the direction at r . Hence, only the φ component of F → will survive. F φ = p m ∂ ∂ r μ 0 I 2 π r = - μ 0 I p m 2 π r 2 . (c) The vector p m → coincides in direction with the magnetic field, which is produced by the conductor carrying current I , that is, the direction of p m → is along the direction of φ ^ . Therefore, F → = p m ∂ r ∂ φ μ 0 I 2 π φ → = μ 0 I p m 2 π r 2 ∂ φ → ∂ φ = μ 0 I p m 2 π r 2 ∂ ( φ φ ^ ) ∂ φ ∵ φ → = φ φ ^ Now, ∂ φ ^ ∂ φ = - e r ^ So, F → = - μ 0 I p m 2 π r 2 e r ^

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What pressure does the lateral surface of a long straight solenoid, with $n$ turns per unit length, experience, when a current $I$ flows through it?

Important Questions on ELECTRODYNAMICS

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Problems in General Physics>Chapter 3 - ELECTRODYNAMICS>CONSTANT MAGNETIC FIELD, MAGNETICS>Q 3.264.

A current

I

flows in a long single-layer solenoid with cross-sectional radius

R

. The number of turns per unit length, of the solenoid, equals

n

. Find the limiting current at which the winding may rupture, if the tensile strength of the wire is equal to

F_{\lim}

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Problems in General Physics>Chapter 3 - ELECTRODYNAMICS>CONSTANT MAGNETIC FIELD, MAGNETICS>Q 3.265.

A parallel-plate capacitor with area of each plate equal to

S

, and the separation between them, to

d

, is put into a stream of conducting liquid with resistivity

ρ

. The liquid moves parallel to the plates with a constant velocity

\vec{v}

. The whole system is located in a uniform magnetic field of induction

\vec{B}

, vector

\vec{B}

being parallel to the plates and perpendicular to the stream direction. The capacitor plates are interconnected by means of an external resistance

R

. What amount of power is generated in that resistance? At what value of

R

, is the generated power the highest? What is this highest power equal to?

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Problems in General Physics>Chapter 3 - ELECTRODYNAMICS>CONSTANT MAGNETIC FIELD, MAGNETICS>Q 3.266.

A straight round copper conductor, of radius

R = 5.0 mm

, carries a current

I = 50

A

. Find the potential difference between the axis of the conductor and its surface. The concentration of the conduction electrons in copper is equal to

n = 0.9 \times 10^{23} {cm}^{- 3}

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Problems in General Physics>Chapter 3 - ELECTRODYNAMICS>CONSTANT MAGNETIC FIELD, MAGNETICS>Q 3.269.

A small current-carrying loop is located at a distance

r

from a long straight conductor with current

I

. The magnetic moment of the loop is equal to

\vec{p_{m}}

. Find the magnitude and direction of the force vector applied to the loop, if

\vec{p_{m}}

(a) is parallel to the straight conductor,
(b) is oriented along the radius vector

\vec{r}

,
(c) coincides in the direction with the magnetic field produced by the current

I

, at the point where the loop is located.

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Problems in General Physics>Chapter 3 - ELECTRODYNAMICS>CONSTANT MAGNETIC FIELD, MAGNETICS>Q 3.270.

A small current-carrying coil, having a magnetic moment

\vec{p_{m}}

, is located at the axis of a round loop, of radius

R

, with current

I

flowing through it. Find the magnitude of the vector force applied to the coil, if its distance from the centre of the loop is equal to

x

, and the vector

\vec{p_{m}}

coincides in the direction with the axis of the loop.

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Problems in General Physics>Chapter 3 - ELECTRODYNAMICS>CONSTANT MAGNETIC FIELD, MAGNETICS>Q 3.271.

Find the interaction force of two coils with magnetic moments

p_{1 m} = 4.0 mA m^{2}

and

p_{2 m} = 6.0 mA m^{2}

, and collinear axes, if the separation between the coils is equal to

l = 20 cm

, which exceeds considerably their linear dimensions. (permeability of vacuum,

μ_{0} = 1.257 \times 10^{- 6} H m^{- 1}

)

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Problems in General Physics>Chapter 3 - ELECTRODYNAMICS>ELECTROMAGNETIC INDUCTION. MAXWELL'S EQUATIONS>Q 3.288.

A wire, bent as a parabola $y = a x^{2}$ , is located in a uniform magnetic field of induction $\vec{B}$ , the vector $\vec{B}$ being perpendicular to the plane $x - y$ . At the moment $t = 0$ , a connector starts sliding translation-wise from the parabola apex, with a constant acceleration $\vec{w}$ . Find the emf of electromagnetic induction in the loop thus formed as a function of $y$ .

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Problems in General Physics>Chapter 3 - ELECTRODYNAMICS>ELECTROMAGNETIC INDUCTION. MAXWELL'S EQUATIONS>Q 3.289.

A rectangular loop with a sliding connector of length $I$ , is located in a uniform magnetic field perpendicular to the loop plane. The magnetic induction is equal to $B$ . The connector has an electric resistance $R$ , the sides $A B$ and $C D$ have resistances $R_{1}$ and $R_{2}$ , respectively. Neglecting the self-inductance of the loop, find the current flowing in the connector during its motion, with a constant velocity $v$ .

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What pressure does the lateral surface of a long straight solenoid, with n turns per unit length, experience, when a current I flows through it?

Important Questions on ELECTRODYNAMICS

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What pressure does the lateral surface of a long straight solenoid, with $n$ turns per unit length, experience, when a current $I$ flows through it?