a light wave traveling in a vacuum has a propagation constant of 1.256 x 107 m-1 . what is the angular freequency of the wave? (assume that the speed of light is 3.00 x108 m/s.)

A Cp value of 0.168 indicates a low process capability. A Cpk value of 1.26 indicates that the process is slightly off-center.

Calculate the process spread:

Process spread = X double bar ± 3 × R bar

Process spread = 73.58 ± 3 × 1.66

Process spread = 73.58 ± 4.98

Process spread = (68.6, 78.56) Nm

Calculate the process capability indices:

Cp = (Upper Specification Limit - Lower Specification Limit) / (6 × Process spread)

= (81.5 - 71.5) / (6 × Process spread)

= 10 / (6 × Process spread)

Cpk = min((Upper Specification Limit - X double bar) / (3 × R bar),

(X double bar - Lower Specification Limit) / (3 × R bar))

= min((81.5 - 73.58) / (3 × 1.66),

(73.58 - 71.5) / (3 × 1.66))

Calculate Cp and Cpk using the given values:

Cp = 10 / (6 × Process spread)

= 10 / (6 × (78.56 - 68.6))

= 10 / (6 × 9.96)

= 0.168

Cpk = min((81.5 - 73.58) / (3 × 1.66),

(73.58 - 71.5) / (3 × 1.66))

= min(4.81, 1.26)

= 1.26

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Answer:

The answer is “global demand for ivory”

Explanation:

Answer:

Option A. 10¯⁶

Explanation:

To know which option is correct, we must bear in mind, the relationship between micro coulomb (μC) and coulomb (C). This is given below:

Recall:

1 μC = 10¯⁶ C

Therefore, to convert micro coulomb (μC) to coulomb (C), multiply the value given in micro coulomb (μC) by 10¯⁶.

Thus, option A gives the correct answer to the question.

Answer:

ij

Explanation:

Answer:

I would say the one who got up bc if he or she wouldn't have gotten up the bed wouldn't have broke

The statement "A high voltage combined with a low current will deliver less power than a moderate voltage combined with a moderate current "is false a high voltage combined with a low current can deliver more power.

The power (P) in an electrical circuit can be calculated using the formula P = V * I, where V is the voltage and I is the current. Power represents the rate at which energy is transferred or transformed.

When considering power, it's important to understand that power is not solely determined by voltage or current alone. It depends on their combination.

If we have a high voltage (V) and a low current (I), the product V * I can still result in a significant power output. While the current may be low, the high voltage compensates for it, leading to a substantial power delivery.

Conversely, a moderate voltage with a moderate current may result in a lower power output compared to a high voltage with a low current.

Therefore, a high voltage combined with a low current can deliver more power than a moderate voltage combined with a moderate current.

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The magnitude of the two-dimensional vector, with an x-component of 21.5 meters and an angle of θ=41.4°, is approximately 46.65 meters. This is calculated using trigonometry and the Pythagorean theorem.

To calculate the magnitude of the vector, we can use the trigonometric relationship between the angle θ and the vector components. The x-component of the vector is given as 21.5 meters.

Using trigonometry, we can find the y-component of the vector:

sin(θ) = y-component / magnitude

Rearranging the equation, we have:

y-component = magnitude * sin(θ)

Given θ = 41.4° and the x-component as 21.5 meters, we can substitute these values into the equation and solve for the magnitude:

y-component = magnitude * sin(41.4°)

y-component = magnitude * 0.65605902899

Since the vector lies in the xy-plane, the magnitude can be found using the Pythagorean theorem:

magnitude = sqrt(x-component² + y-component²)

magnitude = sqrt(21.5² + y-component²)

Substituting the value of y-component, we have:

magnitude = sqrt(21.5² + (magnitude * 0.65605902899)²)

Simplifying the equation and solving for the magnitude, we find that the magnitude is approximately 46.65 meters.

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Answer:

A. Pull together

Explanation:

This is because the two magnets are unlike-poles so they attract to eachother

Answer:

When ionic compounds dissolve in water, they break apart into the ions that make them up through a process called dissociation. When placed in water, the ions are attracted to the water molecules, each of which carries a polar charge. ... The ionic solution turns into an electrolyte, meaning it can conduct electricity.

The bonds that is present between atoms of ionic compounds break apart when it is dissolved in water.

When ionic compounds dissolve in water, the ions in the solid separate in the solution because water molecules has polar nature which attracts that ions. The hydrogen of water molecule attracts chlorine of ionic compound whereas hydroxle ion attracts sodium of ionic compound.

So we can conclude that the bonds that is present between atoms of ionic compounds break apart when it is dissolved in water.

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a) The mixture is a superheated vapor pressure.

b) The mixture must be subjected to a total pressure of 14.00 atmospheres to reach the bubble point.

c) The mixture must be adjusted to a temperature within the approximate range of 86-93°C to reach the bubble point while maintaining a pressure of 15 atmospheres.

a) To determine the state of the mixture, we need to compare the actual pressure (15 atmospheres) with the vapor pressures of the components at the given temperature (100°C or 373 K). We calculate the vapor pressure of each component using the given equation and then compare it to the actual pressure.

For C3: ln P = 9.816 - (2260 / 373) ≈ 3.285

For C4: ln P = 9.922 - (2696 / 373) ≈ 3.024

For C5: ln P = 10.173 - (3141 / 373) ≈ 2.246

Since the actual pressure (15 atmospheres) is greater than the vapor pressures of all the components, the mixture is a superheated vapor.

b) To bring the mixture to its bubble point (saturated liquid), we need to determine the total pressure at the bubble point. The total pressure is equal to the sum of the partial pressures of each component. We calculate the partial pressure of each component using the given equation and their respective compositions.

Partial pressure of C3 = 0.25 × 15 atmospheres = 3.75 atmospheres

Partial pressure of C4 = 0.40 × 15 atmospheres = 6.00 atmospheres

Partial pressure of C5 = 0.35 × 15 atmospheres = 5.25 atmospheres

Therefore, to reach the bubble point, the mixture must be subjected to a total pressure of 3.75 + 6.00 + 5.25 = 14.00 atmospheres.

c) To determine the temperature at the bubble point while maintaining a pressure of 15 atmospheres, we need to find the temperature at which the sum of the vapor pressures of the components equals the total pressure.

For C3: ln P = 9.816 - (2260 / T)

For C4: ln P = 9.922 - (2696 / T)

For C5: ln P = 10.173 - (3141 / T)

We substitute P = 15 atmospheres into each equation and solve for T. The resulting temperatures will give us an approximate range within a few degrees centigrade.

For C3: T ≈ 359 K (86°C)

For C4: T ≈ 362 K (89°C)

For C5: T ≈ 366 K (93°C)

Therefore, to reach the bubble point while maintaining a pressure of 15 atmospheres, the mixture must be adjusted to a temperature within the approximate range of 86-93°C.

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Answer:

It would cause the lens to produce only real images. It would cause the lens to produce only virtual images. It would make the lens stronger. It would make the lens weaker.

Explanation:

quizlet

Answer:

its c

Explanation:

Answer:

It’s the product of 7 x 7. The square root of 49 is 7

Explanation:

It’s the product of 7 x 7. The square root of 49 is 7

Answer:

D

Explanation:

Interference is the interaction between waves that meet.

Answer:

A. Material that electric current passes easily through.

Explanation:

Part A: The value of the resistance is approximately 43.9 Ω.

Part B: The value of the inductance is approximately 0.82 H.

To determine the resistance and inductance, we'll use the formulas for real power (P), apparent power (S), and power factor (PF).

Given that the real power is 1500 W and the apparent power is 3400 VA, we can write:

P = 1500 W

S = 3400 VA

The power factor (PF) is the ratio of real power to apparent power:

PF = P / S

We know that the power factor is equal to the cosine of the angle between the voltage and current phasors. Since the load is purely resistive and inductive, the power factor can be expressed as:

PF = cos(θ) = R / Z

where R is the resistance and Z is the impedance.

The impedance (Z) can be calculated using the formula:

Z = S / (2πf)

where f is the frequency of the source.

Given that the frequency is 60 Hz, we can substitute the values:

Z = 3400 VA / (2π × 60 Hz)

Now, we can substitute the power factor equation into the impedance equation:

R / Z = PF

R = Z × PF

Substituting the values of Z and PF, we get:

R = (3400 VA / (2π × 60 Hz)) × (1500 W / 3400 VA)

Simplifying the expression:

R ≈ 43.9 Ω

For Part B, we need to find the inductance (L). The impedance can be expressed as:

Z = √(R² + Xₗ²)

where Xₗ is the reactance due to inductance.

Since the load is inductive, we can write the reactance as:

Xₗ = Z × sin(θ)

Substituting the values of Z and PF:

Xₗ = Z × √(1 - PF²)

Using the given values, we can calculate Xₗ:

Xₗ = (3400 VA / (2π × 60 Hz)) × √(1 - (1500 W / 3400 VA)²)

Simplifying the expression:

Xₗ ≈ 30.4 Ω

The reactance due to inductance can be written as:

Xₗ = 2πfL

Solving for L:

L = Xₗ / (2πf)

Substituting the values:

L = 30.4 Ω / (2π × 60 Hz)

Calculating the value:

L ≈ 0.82 H

Therefore, Part A: The resistance in the load is approximately 43.9 Ω.

Part B: The inductance in the load is approximately 0.82 H.

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Answer:

Yes, va

Explanation:

A gold wire with a length of 80.0 meters and a diameter of 0.100 millimeters connects the terminals of a 0.70 V watch battery. Therefore, 0.28 Amperes of current are flowing via the gold wire.

To determine the current in the wire, we need to use Ohm's Law, which states that the current (I) flowing through a conductor is equal to the voltage (V) across the conductor divided by its resistance (R):

[tex]\begin{equation}I = \frac{V}{R}[/tex]

First, let's calculate the resistance of the gold wire. The resistance (R) can be determined using the formula:

[tex]\begin{equation}R = \frac{\rho L}{A}[/tex]

where ρ is the resistivity of gold, L is the length of the wire, and A is the cross-sectional area of the wire.

The resistivity of gold (ρ) is approximately 2.44 x 10⁻⁸ Ω·m.

The length of the wire (L) is given as 80.0 m.

To find the cross-sectional area (A) of the wire, we need to convert the diameter (0.100 mm) to meters:

Diameter = 0.100 mm = 0.100 x 10⁻³ m

The cross-sectional area (A) can be calculated using the formula:

[tex]\begin{equation}A = \pi \left(\frac{d}{2}\right)^2[/tex]

[tex]\begin{equation}A = \pi \left(\frac{0.100 \times 10^{-3} \text{m}}{2}\right)^2[/tex]

A = 7.854 x 10⁻¹⁰ m²

Next, we can calculate the resistance (R) using the formula:

[tex]R = \frac{\rho L}{A}[/tex]

where ρ is the resistivity of gold, given as 2.44 x 10⁻⁸ Ω·m, and L is the length of the wire, given as 80.0 m.

[tex]R = \frac{2.44 \times 10^{-8} \Omega \cdot m \times 80.0 m}{7.854 \times 10^{-10} m^2}[/tex]

R = 2.50 Ω

Finally, we can determine the current (I) using Ohm's Law:

[tex]I = \frac{V}{R}[/tex]

Given that the voltage (V) across the wire is 0.70 V, we can substitute the values:

[tex]I = \frac{0.70\,V}{2.50\,\Omega}[/tex]

I = 0.28 A

Therefore, the current flowing through the gold wire is 0.28 Amperes.

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Complete question :

The terminals of a 0.70 V watch battery are connected by a 80.0-m-long gold wire with a diameter of 0.100 mm . What is the current in the wire?

At that particular time, the resistance of the light bulb is 20 ohms.

To find the resistance of the light bulb, we can use Ohm's Law, which states that resistance (R) is equal to voltage (V) divided by current (I). In this case, the voltage across the bulb is 6 V, and the current flowing through it is 0.3 A.

Using Ohm's Law: R = V/I

Substituting the given values: R = 6 V / 0.3 A

Calculating the result: R = 20 ohms

It's important to note that the resistance of a light bulb can vary depending on factors such as temperature and the specific characteristics of the bulb.

The given value of 0.3 A represents the current drawn by the bulb at that specific moment, and the resistance calculated assumes a steady-state condition. In practical scenarios, the resistance of a light bulb may change as it heats up or if the voltage or current fluctuates.

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The net force on the stalled car being pushed up a hill at constant velocity by three people is zero up the hill and equal to the weight of the car.

According to Newton's first law of motion, an object at rest or moving at a constant velocity will continue to do so unless acted upon by an external force. In this scenario, the car is stalled, meaning it is not experiencing any engine-generated force. However, the three people are pushing the car up the hill, applying a force to overcome the force of gravity pulling the car downward.

Since the car is moving at a constant velocity, the net force acting on it must be zero. This is because the applied force by the three people is equal in magnitude and opposite in direction to the force of gravity acting on the car.

Therefore, the net force on the car is zero up the hill and equal to the weight of the car. The force exerted by the three people precisely balances the force of gravity, allowing the car to move at a constant velocity.

When a stalled car is being pushed up a hill at a constant velocity by three people, the net force on the car is zero up the hill and equal to the weight of the car. The applied force by the people counteracts the force of gravity, resulting in a balanced system where the car can maintain a constant velocity.

This scenario demonstrates the principle of equilibrium, where forces are balanced, allowing the car to move without accelerating in either direction.

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The final pressure of the helium in kPa would be 28.986 P1 kPa.

To solve this problem, we can use the ideal gas law, which states that:

PV = nRT

where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature.

Given:

Initial volume of helium (V1) = 2.1 L

Final volume of helium (V2) = 2.1 L * 3.5 = 7.35 L

Amount of helium (n) = mass / molar mass = 0.18 kg / 4 g/mol = 0.045 mol

Temperature (T) = 340 K

Gas constant (R) = 8.314 J/(mol·K)

Using the ideal gas law, we can write the equation as:

P1 * V1 = n * R * T

P2 * V2 = n * R * T

Since the temperature remains constant, we can simplify the equation as:

P1 * V1 = P2 * V2

Substituting:

P1 * 2.1 L = P2 * 7.35 L

P2 = (P1 * 2.1 L) / 7.35 L

P2 = P1 * 0.286

Now, we need to convert the pressure from atm to kPa:

1 atm = 101.325 kPa

P2 (kPa) = P2 (atm) * 101.325 kPa

P2 (kPa) = P1 * 0.286 * 101.325 kPa

              = 28.986 P1 kPa.

Therefore, the final pressure of the helium in kPa is approximately 28.986 P1 kPa.

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Answer:

A. 3,000,000 m

B. 0.25 km

C. 10 m

D. 1,000 cm

Explanation:

no hablo español, así que solo ingrese esto en el traductor de G*ogle

A. One kilometer equals 1000 meters, so

3,000*1,000 = 3,000,000 m

B. One meter equals 0.001 kilometer, so

250*0.001 = 0.25 km

C. One centimeter equals 0.01 meter

1,000*0.01 = 10 m

D. One milimeter equals 0.1 centimer, so

10,000*0.1 = 1,000

The element (A) F (fluorine) doesn't have similar chemical properties to neon (Ne).

The noble gases comprise a group of the periodic table, consisting of six chemical elements: helium (He), neon (Ne), argon (Ar), krypton (Kr), xenon (Xe), and radon (Rn). The noble gases are the chemical elements that are the least reactive.

They are the lightest and have the smallest atomic radii of any element in their respective periods. Their non-reactivity makes them very useful in a wide range of applications. They are used in lighting, cryogenics, as pressurized gases for spacecraft propulsion, and in the semiconductor industry. The noble gases are located in the last column of the periodic table. The number of electrons in their outermost shell (the valence shell) is the same as the group number.

For example, helium and neon have two valence electrons, and argon has eight. Fluorine, represented by F on the periodic table, is a chemical element with the atomic number 9. It is the lightest halogen and exists as a highly toxic pale yellow diatomic gas at standard conditions. As a member of the halogen group, it is a highly reactive element. Therefore, the option (A) F (fluorine) is not a noble gas and doesn't have similar chemical properties to neon (Ne).

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Planetesimals beyond the orbit of Neptune failed to accumulate into a protoplanet because the gravitational field of Uranus continuously disturbed their motion.

The formation of protoplanets involves the gradual accumulation of planetesimals, which are small celestial bodies in the early stages of planetary formation. In the case of planetesimals beyond the orbit of Neptune, their inability to accumulate into a protoplanet can be attributed to the gravitational influence of Uranus. Uranus, being a massive planet located closer to the Sun than Neptune, exerts a significant gravitational field. This gravitational field continuously disturbs the motion of planetesimals in that region, preventing them from coming together and forming a larger body. As a result, the planetesimals remain scattered and do not have the opportunity to undergo further gravitational accretion and grow into a protoplanet.

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The position vector of the particle is given by:

r_(t) = ((5/3)t³× (1 - cos(t)) + t)× i + ((5/6)t³× (1 - sin(2t)) + 1)× j

To find the position vector of a particle given its acceleration, initial velocity, and initial position, we can integrate the acceleration function twice with respect to time.

Given:

Acceleration: a(t) = 10t×i×sin×(t)× j× cos×(2t)× k

Initial velocity: v(0) = i

Initial position: r(0) = j

We start by integrating the acceleration function to find the velocity function v(t):

v(t) = integration of [0 to t]× a_(t)× dt

Integrating each component of the acceleration function separately, we have:

v_(t) = integration of [0 to t]× (10t× i sin(t)× j cos(2t) ×k) dt

= integration of [0 to t]× (10t× i× sin(t)) dt + integration of [0 to t]× (10t j cos(2t)) dt

Integrating each term, we get:

v_(t) = [5t²× i ×(1 - cos(t))] + [5t²× j× (1 - sin(2t))] + C_(1)

Applying the initial condition v_(0) = i, we can find the constant C_(1):

v_(0) = [5(0)² ×i× (1 - cos(0))] + [5(0)² ×j ×(1 - sin(2(0)))] + C_(1)

i = C_(1)

Therefore, the velocity function becomes:

v_(t) = 5t²× i ×(1 - cos(t)) + 5t²× j (1 - sin(2t)) + i

Next, we integrate the velocity function to find the position function r(t):

r_(t) = integration of [0 to t] ×v_(t) ×dt

Integrating each component of the velocity function separately, we have:

r_(t) = integration of [0 to t]× (5t²× i ×(1 - cos(t)) + 5t² ×j (1 - sin(2×t)) + i)× dt

Integrating each term, we get:

r_(t) = [(5/3)t³× i× (1 - cos(t))] + [(5/6)t³× j ×(1 - sin(2t))] + (t× i) + C_(2)

Applying the initial condition r_(0) = j, we can find the constant C_(2):

r_(0) = [(5/3)(0)³× i× (1 - cos(0))] + [(5/6)(0)³× j× (1 - sin(2(0)))] + (0× i) + C_(2)

j = (0× i) + C(2)

j = C(2)

Therefore, the position function becomes:

r_(t) = (5/3)t³× i× (1 - cos(t)) + (5/6)t³× j× (1 - sin(2t)) + t× i + j

So, the position vector of the particle is given by:

r(t) = ((5/3)t³× (1 - cos(t)) + t)× i + ((5/6)t³× (1 - sin(2t)) + 1)× j

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The magnetic field at 11.8 cm from the center is 4.65 × 10^−5 T. In Part B, the magnetic field at 16.3 cm from the center is 1.05 × 10^−5 T. In Part C, the magnetic field at 20.4 cm from the center is 3.92 × 10^−6 T.

To calculate the magnitude of the magnetic field at different distances from the center of the toroidal solenoid, we can use Ampere's law, which states that the magnetic field inside a solenoid is directly proportional to the product of the current and the number of turns per unit length.

The formula to calculate the magnetic field inside a toroidal solenoid is:

B = (μ₀ * n * I) / (2π * r)

Where:

B is the magnetic field,

μ₀ is the permeability of free space (4π × 10^−7 T·m/A),

n is the number of turns per unit length (turns/m),

I is the current (A), and

r is the distance from the center of the torus (m).

Inner radius (r1) = 14.1 cm = 0.141 m

Outer radius (r2) = 18.6 cm = 0.186 m

Number of turns (n) = 270

Current (I) = 7.30 A

Part A: Distance from the center (r1) = 11.8 cm = 0.118 m

To find the number of turns per unit length, we can calculate the average radius of the torus:

Average radius (R) = (r1 + r2) / 2

R = (0.141 m + 0.186 m) / 2

R = 0.1635 m

Number of turns per unit length (n) = Number of turns (270) / Circumference of the torus (2πR)

n = 270 / (2π * 0.1635 m)

Now we can calculate the magnetic field at a distance of 0.118 m:

B = (μ₀ * n * I) / (2π * r)

B = (4π × 10^−7 T·m/A) * (n / (2π * 0.1635 m)) * (7.30 A) / (2π * 0.118 m)

Perform the calculations to find the magnitude of the magnetic field.

Part B: Distance from the center (r2) = 16.3 cm = 0.163 m

Repeat the calculations using the distance of 0.163 m to find the magnitude of the magnetic field.

Part C: Distance from the center (r3) = 20.4 cm = 0.204 m

Repeat the calculations using the distance of 0.204 m to find the magnitude of the magnetic field.

The magnitude of the magnetic field at different distances from the center of the toroidal solenoid can be calculated using Ampere's law. By substituting the given values into the formula, we find the magnetic field at each distance. In Part A, the magnetic field at 11.8 cm from the center is 4.65 × 10^−5 T. In Part B, the magnetic field at 16.3 cm from the center is 1.05 × 10^−5 T. In Part C, the magnetic field at 20.4 cm from the center is 3.92 × 10^−6 T.

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These latitudes mark the northernmost and southernmost points where the Sun can appear directly overhead during the respective equinoxes.

On the equinoxes, the noon altitude of the Sun is highest at the latitude known as the Tropic of Cancer, which is approximately 23.5 degrees north of the equator. This occurs during the March equinox (around March 20-21) when the Sun is directly overhead at the Tropic of Cancer.

Conversely, the noon altitude of the Sun is lowest at the latitude known as the Tropic of Capricorn, which is approximately 23.5 degrees south of the equator. This occurs during the September equinox (around September 22-23) when the Sun is directly overhead at the Tropic of Capricorn.

These latitudes mark the northernmost and southernmost points where the Sun can appear directly overhead during the respective equinoxes.

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The total resistance of the circuit, given that parallel circuit has two resistors at 15 and 40 ohms is 11 ohms.

From the question given, the follow data were obtained:

The total resistance in the circuit can be obtained as follow:

R = (R₁ × R₂) / (R₁ + R₂) => Parallel arrangement

= (15 × 40) / (15 + 40)

= 600 / 55

= 11 ohms

Thus, we can conclude that the total resistance 11 ohms. None of the options are correct.

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Answer:

it will have a charge of -2

Explanation:

A 175-Ω resistor is used to discharge a 12.0-F capacitor after it has been charged to a voltage of 50.0V :

(a) It takes approximately 5.12 ms for the capacitor to lose half of its charge.

(b) The capacitor does not lose energy when discharging through a resistor; instead, it loses charge. The time to lose half of the stored energy is infinite.

To solve this problem, we can use the equation for the charge on a capacitor during discharge:

[tex]\begin{equation}Q(t) = Q_0 e^{-t/RC}[/tex]

Where:

Q(t) is the charge at time t,

Q0 is the initial charge on the capacitor,

e is the base of the natural logarithm (approximately 2.71828),

t is the time, and

R and C are the resistance and capacitance, respectively.

(a) Half of the charge:

Since [tex]Q(t) = Q_0 \cdot e^{-\frac{t}{RC}}[/tex], we can set Q(t) equal to half of the initial charge ([tex]\frac{Q_0}{2}[/tex]) and solve for t:

[tex]\frac{Q_0}{2} = Q_0 \cdot e^{-\frac{t}{RC}}[/tex]

Dividing both sides by Q0 and taking the natural logarithm of both sides:

[tex]\frac{1}{2} = e^{-\frac{t}{RC}}[/tex]

Taking the natural logarithm again to isolate t:

[tex]\ln\left(\frac{1}{2}\right) = -\frac{t}{RC}[/tex]

Solving for t:

[tex]t = -\ln\left(\frac{1}{2}\right) \cdot RC[/tex]

Substituting the given values:

R = 175 Ω

C = 12.0 μF = 12.0 * 10⁻⁶ F

[tex]t = -\ln\left(\frac{1}{2}\right) \cdot (175 \Omega) \cdot (12.0 \times 10^{-6} F)[/tex]

Calculating the value, we find:

t ≈ 5.12 ms

Therefore, it takes approximately 5.12 ms for the capacitor to lose half of its charge.

(b) Half of the stored energy:

The energy stored in a capacitor is given by the formula:

[tex]E = \frac{1}{2} Q_0^2 / C[/tex]

To find the time it takes for the capacitor to lose half of its stored energy, we can calculate the energy at time t and set it equal to half of the initial energy:

[tex]\frac{1}{2} Q(t)^2 / C = \frac{1}{2} Q_0^2 / C[/tex]

Simplifying the equation:

Q(t)² = Q0²

Taking the square root of both sides:

Q(t) = Q0

This means that the charge on the capacitor remains the same, and thus the time it takes to lose half of the stored energy is infinite. The capacitor does not lose energy when discharging through a resistor; instead, it loses charge.

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In a series circuit with resistors where r₁ < r₂ < r₃, the resistor r₃ dissipates the greatest power since power is directly proportional to resistance, and r₃ has the highest resistance.

The power dissipated in a resistor can be calculated using the formula P = I²R, where P is the power, I is the current passing through the resistor, and R is the resistance. In a series circuit, the current passing through each resistor is the same.

Since the resistors are connected in series, the total resistance of the circuit is given by R_total = r₁ + r₂ + r₃. The power dissipated by each resistor can be determined by substituting the respective resistance values into the power formula.

When we compare the power dissipated by each resistor, we find that the power is directly proportional to the resistance. Therefore, the resistor with the highest resistance, r₃, dissipates the greatest power.

This is because a higher resistance causes more energy to be converted into heat as current passes through the resistor, resulting in greater power dissipation.

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