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physics
the physics energy
The Physics of Energy 1st edition Robert L. Jaffe, Washington Taylor - Solutions
Design a transmission system to carry power from wind farms in North Dakota to the state of Illinois (about 1200 km). The system should handle Illinois’s summer time electricity demand of 42GW. Land for transmission towers is at a premium, so the system uses very high voltage (VRMS = 765
Prove that the mutual inductance is a symmetric relation, M12 = M21, by computing the energy stored when current I1 is established in loop 1 and then current I2 is established in loop 2. Then set up the same currents in the opposite order.
Consider a long, hollow solenoid of volume V.Show that, ignoring end effects, its inductance is L = n2Vμ0, and that the magnetic energy it stores EEM = LI2/2 can be written in the form of eq. d³x \B(x)? 2µ EEM
Explain why the integral that appears in eq. is independent of the choice of surface S .[make use of eq. k dS · B dS (V x E) = dt dS B
Derived the EMF on a wire loop rotating in a magnetic field using the Lorentz force law to compute the forces on the mobile charges. Although Faraday?s law of induction does not apply in a rotating reference frame, show that the same relation follows from Faraday?s law. F dx wl Bò sine.
Consider the motor. If the resistance in the wire wrapping the rotor is 1Ω, compute the energy lost under the conditions described. What fraction of energy is lost to Joule heating in this situation?If the current is doubled but the rotation rate is kept fixed, how do the output power, Joule
An electric motor operates at 1000 rpm with an average torque of τ = 0.1Nm. What is its power output? If it is running on 1.2A of current, estimate the back-EMF from the rotor.
Derive eq. (3.52) from eq. (3.43) and eq. (3.50).Make sure you get the both the direction and magnitude. dF (x) = 1 dx x B(x),
Show that the force per unit area on the windings of an air-core solenoid from the magnetic field of the solenoid itself is of order F/A ∼ B2/μ0. Check that the dimensions of this expression are correct and estimate F/A in pascals if |B| = 1T.
Review how magnetic fields are calculated from Ampere?s law by computing? (a) the magnetic field due to a straight wire? (b) the magnetic field in the interior of a very long solenoid.? The contours C1, C2 shown in red in Figure 3.18 will help. You may use the fact that the magnetic field
Show that a charged particle moving in the xy-plane in the presence of a magnetic field B = Bẑ will move in a circle. Compute the radius of the circle and frequency of rotation in terms of the speed v of the particle. Show that adding a constant velocity uẑ in the z-direction stillgives a
Use the magnetic force law and the definition of work? to show that magnetic forces do no work.? W = - dx · F la
Electrical power is often used to boil water for cooking.Here are the results of an experiment: a liter of water initially at 30 ◦C was boiled on an electric stove top burner. The burner is rated at 6.67A (maximum instantaneous current) and 240 VRMS. It took 7 minutes 40 seconds to reach the
An appliance that uses 1000W of power is connected by 12 gauge (diameter 2.053mm) copper wire to a120V (RMS) AC outlet. Estimate the power lost permeter, dPlost/dL, (in W/m) as resistive heating in the wire. (Remember that the wire to the appliance has two separate current-carrying wires in a
Consider two resistors placed in series, one after the other, in an electric circuit connected to a battery with voltage V. Show that the effective resistance of the pair is R1+R2 by using the fact that the current through both resistors is the same while voltages add. Now connect the resistors in
If each of the batteries used in the flashlight has an internal resistance of 0.5Ω (in series with the circuit), what fraction of power is lost to Joule heating within the batteries?
Consider an electric dipole composed of two charges ±Q at positions ± ξ/2. Write the exact electric field from the two charges and show that the leading term in an expansion in 1/r matches the E-field quoted.
A cloud-to-ground lightning bolt can be modeled as a parallel plate capacitor discharge, with Earth’s surface and the bottom of the cloud forming the two plates. A particular bolt of lightning passes to the ground from a cloud bottom at a height of 300 m. The bolt transfers a total charge of 5C
The dielectrics in capacitors allow some leakage current to pass from one plate to the other. The leakage can be parameterized in terms of a leakage resistance RL. This limits the amount of time a capacitor can be used to store electromagnetic energy. The circuit diagram describing an isolated
Suppose that a capacitor with capacitance C is charged to some voltage V and then allowed to discharge through a resistance R. Write an equation governing the rate at which energy in the capacitor decreases with time due to resistive heating. Show that the solution of this equation is EEM(t) =
Starting from Gauss?s law and ignoring edge effects(i.e. assume that the plates are very large and the electric field is uniform and perpendicular to the plates),derive the formula for the capacitance of a parallel plate capacitor, C =??0A/d. Refer to Figure 3.8 ? You can assume that the electric
How much energy can you store on a parallel plate capacitor with d = 1 μm, A = 10 cm2, and ε = 100 ε0,assuming that the breakdown field of the dielectric is the same as for air?
Prove Gauss’s law from Coulomb’s law for static charge distributions by showing that the electric field of a single charge satisfies the integral form of Gauss’slaw and then invoking linearity.
The electric field outside a charged conducting sphere is the same as if the charge were centered at its origin.Use this fact to calculate the capacitance of a sphere of radius R, taking the second conductor to be located at infinity. What is the most charge you can store on an otherwise isolated
In the American game of baseball, a pitcher throws a baseball, which is a round sphere of diameter b = 0.075 m, a distance of 18.4 m (60.5 feet), to a batter, who tries to hit the ball as far as he can. A baseball has a mass close to 0.15 kg. A radar gun measures the speed of a baseball at the time
Estimate Earth’s kinetic energy of rotation (the moment of inertia of a uniform sphere is 2/5 MR2).
Verify the assertion (see example 2.3) that Ekin =?1/2 V for the Moon in a circular orbit around Earth. [The magnitude of the centripetal acceleration for circular motion?can be rewritten a = v2/r.] a = i = -a*r
Estimate the rotational kinetic energy in a spinning yoyo (a plastic toy that you can assume is a cylinder of diameter 5.7 cm and mass 52 g, which rotates at 100 Hz). Compare to the gravitational potential energy of the yo-yo at height of 0.75 m.
If the vehicle used as an example in this chapter accelerates to 50 km/h between each stop light, find the maximum distance between stoplights for which the energy used to accelerate the vehicle exceeds the energy lost to air resistance. (May assume that the time for acceleration is negligible in
One way to estimate the effective area of an object is to measure its limiting velocity v∞ falling in air. Explain how this works and find the expression for Aeff as a function of m (the mass of the object), v∞, g, and the density of air. The drag coefficient of a soccer ball (radius
Consider an idealized cylinder of cross-sectional area A moving along its axis through an idealized diffuse gas of air molecules with vanishing initial velocity. Assume that the air molecules are point like and do not interact with one another. Compute the velocity that each air molecule acquires
Compare the rate of power lost to air resistance for the following two vehicles at 60 km/h and 120 km/h: (a) General Motors EV1 with cdA ≅ 0.37m2, (b) Hummer H2 with cdA ≅ 2.45m2.
Estimate the power output of an elite cyclist pedaling a bicycle on a flat road at 14 m/s. Assume all energy is lost to air resistance, the cross-sectional area of the cyclist and the bicycle is 0.4 m2, and the drag coefficient is 0.75. Now estimate the power output of the same elite cyclist
Consider an airplane with mass 70 000 kg, cross sectional area 12 m2, and drag coefficient 0.03. Estimate the energy needed to get the plane moving at 800 km/h and lift the plane to 10000 m, and estimate air-resistance losses for a flight of 2000 km using the formula in the previous problem. Do a
The density of air decreases with altitude roughly as ρ ∼ ρ0e−z/H, where z is the height above the Earth’s surface and H ≅8.5 km near the surface. Compute the ratio of air resistance losses for an airplane traveling at 750 km/h at an altitude of 12000 m compared to an altitude of 2000 m.
Consider a collision between two objects of different masses M,m moving in one dimension with initial speeds V, −v. In the center-of-mass frame, the total momentum is zero before and after the collision. In an elastic collision both energy and momentum are conserved. Compute the velocities of the
Verify the claim that conversion of the potential energy at the top of a 15 m hill to kinetic energy would increase the speed of the Toyota Camry from 62 to 74 mph.
Relate eq.(2.20)?? to eq. (2.9) and compute g in terms of G, M?, R? as described in the text. GMm V (r) =
Estimate the potential energy of the international space station (mass 370000 kg) relative to Earth’s surface when in orbit at a height of 350 km. Compute the velocity of the space station in a circular orbit and compare the kinetic and potential energy.
An object of mass m is held near the origin by a spring, F = −kx. What is its potential energy? Show that x(t) = (x0, y0, z0) cos ωt is a possible classical motion for the mass. What is the energy of this solution? Is this the most general motion? If not, give an example of another solution.
A mass m moves under the influence of a force derived from the potential V(x) = V0 cosh ax , where the properties of cosh x, What is the force on the mass? What is the frequency of small oscillations about the origin? As the magnitude of the oscillations grows, does their frequency increase,
Estimate the kinetic energy of the ocean liner Queen Mary 2, with mass (displacement) 76000 tons, moving at a cruising speed of 26 knots. Compare with the change in potential energy of the Queen Mary 2 when lifted by a three foot tide.
Use any means at your disposal (including the internet) to justify estimates of the following: (a) The kinetic energy of the Moon in its motion around Earth. (b) The kinetic energy of a raindrop before it hits the ground. (c) The potential energy of the water in Lake Powell (relative
Choose your favorite local mountain. Estimate how much energy it takes to hike to the top of the mountain (ignoring all the local ups and downs of a typical trail). How does your result compare to a typical day’s food energy intake of 10 MJ, taking into account the fact that human muscles are not
Make a rough estimate of the maximum hydropower available from rainfall in the US state of Colorado. Look up the average yearly rainfall and average elevation of Colorado and estimate the potential energy (with respect to sea level) of all the water that falls on Colorado over a year. How does this
The US total electrical power consumption in 2010 was 3.9 TkWh. Utilities try to maintain a capacity that is twice the average power consumption to allow for high demand on hot summer days. What installed generating capacity does this imply?
The energy available from one kilogram of 235U is 82 TJ. Energy is released when each uranium nucleus splits, or fissions. (The fission process is described in §16, but you do not need to know anything about the process for this problem.) 235 grams of 235U contain approximately Avagadro’s
The energy emitted or absorbed in chemical processes is often quoted in kilo joules per mole (abbreviated mol) of reactants, where a mole contains NA ≅ 6.022 x 1023(Avogadro’s number) molecules (§5). Derive the conversion factor from eV/molecule to kJ/mol.
The gravitational potential energy of an object of mass m at a distance h above ground is given by E = mgh . Use dimensional analysis to compute the units of g. What does this result suggest about the behavior of objects near Earth’s surface?
The US total energy consumption in 2014 was 98.0 quads. What is this quantity in joules? About 83% of US energy comes from fossil fuels. If the whole 83% came from oil, how many tons of oil equivalent (toe) would this have been? How many barrels of oil (bbl) is this equivalent to?
In a typical mid-latitude location, incident solar energy averages around 200 W/m2 over a 24-hour cycle. Compute the land area needed to supply the average person’s energy use of 200 MJ/day if solar panels convert a net 5% of the energy incident over a large area of land to useful electrical
How much energy would a 100 W light bulb consume if left on for 10 hours?
Confirm eq.and the estimate for the rhinoceros’s kinetic energy below eq. by explicit calculation. Ep+p=2MPC² = 3 x 10-10 JE 1877 MeV. kinetic energy = mv²
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