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Magnetism

Consider a region of space free from matter and from magnetic fields.Show that it cannot be permeated by an electric field of the form:

\vec{E}(x, y, z)=\frac{A}{2}(y \hat{x}-x \hat{y}+z \hat{z})

where A is a constant value different from zero and with units V/m2.

Now, consider that a magnetic field permeates the region of(a). Write an expression representing a time-varying magnetic field B(r,y, z, t) that renders the vector field proposed in (a) suitable to describe an electric field. Check and demonstrate that the expression you found is a suitable representation of a magnetic field.

Hence or otherwise, show that:

\vec{B}=K x \hat{x}+K y \hat{y}+A t \hat{z}

is not an adequate expression for a magnetic field for K # 0 T/m2.

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Magnetism

Use macroscopic Ohm's and Kirchhoff's laws to demonstrate that thetotal resistance of two resistors connected in series in a eircuit equatesthe sum of their resistance.[6]

Use macroscopic Ohm's and Kirchhoff's laws to demonstrate that thetotal resistance R of two resistors of resistance R and R2 connected 5]in parallel is R = (R'+ R').

Calculate the total capacitance C that two capacitors with capacitanceC1 = 1 pF and C, = 2 pF introduce in a circuit if they are connected in series.

Calculate the total capacitance C that two capacitors with capaci-tance C = 11 pF and C = 22 p F introduce in a circuit if they are connected in parallel.

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Magnetism

A hollow spherical conductor of internal radius R2 and external radius R3surrounds a conductive sphere of radius R1, which is charged with a charge Q,as shown in Figure Q1.

Derive the expression for the magnitude of the electric field E(r), for between 0 and infinity. Note that r =reference system, as shown in Figure 01.0 is the origin of the spherical

Hence or otherwise derive the expression for the scalar potential v(r) for r between 0 and infinity

Give a qualitative graphical representation for the functions E(r)(magnitude of the electric field) and V(r) (scalar potential), consid-ering R1 = 30 cm, R2 = 50 cm, R3 = 80 cm, and QUse the appropriate units on the graphs. Write the values of the elec-tric field in the dielectric side of the interface for each of the metallic800 x 10-12 C.%3Dsurfaces. Write the values for the potential at r = 0, r = R1, r = R2,and r = R3.

Briefly describe what happens to the scalar potential and electric field in the region with R <r < R2 at the static equilibrium if a charged sphere of radius R,close to the hollow conductor, with its centre at R = 4 m. Include a brief explanation for your answer.= 20 cm and charge Q, = 10-6 C is positioned

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Magnetism

. A simple atom has a ground state |g) and an excited state |e), with energies E, :E = E, respectively.= 0 and

(a) Draw the energy level diagram, and label all relevant aspects.

(b) The atom is prepared in the state

|\psi\rangle=\sqrt{\frac{2}{3}}|g\rangle+\sqrt{\frac{1}{3}}|e\rangle

Calculate the probability of finding the atom in the excited state Je).

(c) Calculate the expectation value for the energy of the atom in state psi

The time evolution of the atom is governed by the Hamiltonian H, with

H=E_{g}|g\rangle\left\langle g\left|+E_{e}\right| e\right\rangle\langle e|

Calculate the state of the atom at time t = T, given that the atom is in state |) at time t = 0.

At time T we measure whether the atom is in the state

|+\rangle=\frac{|g\rangle+|e\rangle}{\sqrt{2}} \quad \text { or } \quad|-\rangle=\frac{|g\rangle-|e\rangle}{\sqrt{2}}

Calculate the probability of finding the atom in state |+), and sketch this probability as a function of time.

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Magnetism

(a) The temperature coefficient of resistivity of copper is 3.9 x 103K. The resistivity of copper1 mm is heated to 80°C.1.72 x 10-8 m at 20°C. A copper wire of length 10 cm and diameter

i. Calculate the resistivity of copper at 80°C.

ii. The wire is connected to a battery providing a voltage of 1.5 V. Calculate the power dissipated in the wire.

A sphere of radius R carries a charge density

\rho(r)=\frac{5 Q r^{2}}{4 \pi R^{5}}

where r is the radial distance from the centre of the sphere.

i. Sketch the charge densityas a function of r from r = 0 to r = 2R.

ii. Show that the total charge of the sphere is Q.

iii. Calculate the electric field E inside the sphere.

iv. Calculate the electric field outside the sphere.

v. Sketch the electric field magnitude as a function of r from r = 0 to r = 2R.

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Magnetism

. A circular coil of radius a with N turns lies in the xy plane with the z axis through itscentre, as shown in Fig. 1. The magnetic field along the axis is given by:

B(z)=\frac{\mu_{0} N I a^{2}}{2\left(a^{2}+z^{2}\right)^{3 / 2}}

:0.20 A, andN =5.0 x 10-Am2 lies along the z axis at a distance of zwhere I is the current. The coil has a =1.0 cm, I =1000. A magneticdipole with magnitude m =+5.0 cm from the centre of the coil. The dipole points along the +z axis.

(a) What is the torque on the dipole?

(b) What is the magnetic energy of the dipole?

(c) What is the force on the dipole? (Hint: make the approximation z? > a².) Byconsidering the coil as a dipole, and making the analogy with bar-magnet dipoles,explain the sign of the force on the dipole.

(d) Sketch the dipole's magnetic energy as a function of z, and describe its motion, as-suming that it is free to move without any frictional forces. (Hint: make an analogywith a ball rolling on a curved surface, and apply conservation of energy.)

(e) The dipole has a mass of 7.9 x 10-6 kg. What is its maximum speed?

(f) The dipole is made of ferromagnetic iron, which has a relative atomic mass of 55.8.Calculate the average dipole moment per iron atom along the z axis in units of theBohr magneton, UB. Explain how this value can be significantly less than uB, eventhough each individual iron atom has a dipole moment of - pg.

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Magnetism

4. (5 point) An infinitely-long cylindrical wire with radius a is made of perfect conductor and is located above a perfect conducting ground plane. If the distance between the center of the wire and the ground plane is 2a, then what is the unit capacitance between the wire and the ground?

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Magnetism

5. An electron moving at 2.0x10' ms passes at 90° through a magnetic field of flux density 2.5x10² T.The electron has a mass of 9.1x1031 kg, and a charge of -1.6x1019 C.

a) What force is exerted on the electron by the field?

b) What is the radius of the electron's circular path in the field?

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Magnetism

What is the Magnetic field at (a) 20 mm and (b) 40 mm from the wire?

4.A vertical wire carries a current of 6.0A

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Magnetism

3. What rule is used to determine the direction of a magnetic field in a current carrying wire. Illustrate using a diagram.(3)

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