posted 10 months ago

\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.

posted 10 months ago

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.

posted 10 months ago

Briefly explain why a potential difference AV = V - V, arises at the equilibrium between the left and right sides of the conductor, as shown in the figure.

Starting from J = n, qữa and considering the Lorentz force, derive the force acting on the carriers responsible for the current I function of the relevant parameters a, b, I, B and the carrier density ng: Use such result to demonstrate that the carriers responsible for the current in the metal are negatively charged if the potential difference measured between the left and the right facets of the conductor is positive:deltaV = V - V, > 0. To this end, consider that V- V, is positive if the carriers accumulated on the right side have a negative charge.

Suppose that the carriers are electrons ( -1.6x 10-19 C) and thatAV = V,- V, = 22 x 10-6 V. Caleulate the carrier density ngmetal, expressed in units of m .%3Din the

Hence or otherwise, suppose that the current I is induced by a generator keeping the potential difference between the input and and output facets (sand S) at a constant value DElta Vt/0=0.002 V what is the resistivity p of the conductor

posted 10 months ago

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

posted 11 months ago

posted 11 months ago

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.

posted 11 months ago

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.

posted 11 months ago

(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.

posted 1 years ago

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?

posted 1 years ago

4.A vertical wire carries a current of 6.0A