The current-density magnitude in a certain circular wire is J = (2.75 × 10
10
⁄ )r
2
,
where r is the radial distance out to the wire’s radius of 3.00 mm. The potential applied to
the wire (end to end) is 60.0 V. How much energy is converted to thermal energy in
2.00 h?
k=2.75•10¹º A/m⁴ R=0.003 m
i=∫JdA=∫k•r²•2•π•r•dr= k•π•R²/2 =3.5 A
The rate of thermal energy generation is P=i•V=3.5•60=210 W.
Assuming a steady rate, the thermal energy generated in 2h=7200 s is
Q=P •Δt=210•7200= 1.5112 •10⁶ J
Well, let me calculate that for you. But before I do, let me ask you this - why did the resistor bring a flashlight to the party?
Because it wanted to be a "conductor" of the light! Ba-dum-tss!
Alright, let's get cracking with the calculation. We can use the formula for power in terms of current and potential difference: P = IV. Since we have the current density, we can relate it to the current by I = JA, where A is the cross-sectional area of the wire.
The cross-sectional area of a circular wire is given by A = πr^2. Plugging in the values, we have A = π(0.003 m)^2 = 2.83 x 10^-5 m^2.
Therefore, I = (2.75 x 10^10 A/m^2) × (2.83 x 10^-5 m^2) = 778.25 A.
Now we can calculate the power using P = IV: P = (778.25 A) × (60.0 V) = 46695 W.
Since energy (E) is equal to power (P) multiplied by time (t), we have E = Pt.
Plugging in the values, we get E = (46695 W) × (2.00 h) = 93390 Wh.
So, approximately 93390 watt-hours of energy would be converted to thermal energy in 2.00 hours.
To calculate the amount of energy converted to thermal energy in 2.00 hours, we need to determine the power dissipated by the wire and then multiply it by the time.
The power dissipated by the wire can be calculated using the formula: P = I^2 * R, where I is the current and R is the resistance.
First, let's calculate the current flowing through the wire.
Given:
Current-density magnitude, J = (2.75 × 10^10) * r^2
Radial distance, r = 3.00 mm
The current can be found by integrating the current-density magnitude over the cross-sectional area of the wire.
I = ∫ J * dA
Since the wire is circular, the cross-sectional area can be expressed as A = π * r^2.
I = ∫ (2.75 × 10^10) * r^2 * π * r^2 * dr
= (2.75 × 10^10) * π * ∫ r^4 * dr
= (2.75 × 10^10) * π * [r^5 / 5] evaluated from 0 to 3.00 mm
Converting the radial distance to meters,
r = 3.00 mm = 3.00 × 10^-3 m
I = (2.75 × 10^10) * π * [(3.00 × 10^-3)^5 / 5]
≈ 4.86 A (rounded to two decimal places)
Next, we need to calculate the resistance of the wire. The resistance can be calculated using Ohm's Law: R = V / I, where V is the potential applied to the wire.
Given:
Potential applied to the wire, V = 60.0 V
R = V / I
= 60.0 V / 4.86 A
≈ 12.35 Ω (rounded to two decimal places)
Now we can calculate the power dissipated by the wire using the formula: P = I^2 * R.
P = (4.86 A)^2 * 12.35 Ω
≈ 291.63 W (rounded to two decimal places)
Finally, we can calculate the energy converted to thermal energy using the formula: E = P * t, where t is the time.
Given:
Time, t = 2.00 h
Converting the time to seconds,
t = 2.00 h * 60 min/h * 60 s/min
= 7,200 s
E = 291.63 W * 7,200 s
≈ 2,100,000 J (rounded to two decimal places)
Therefore, approximately 2,100,000 joules (J) of energy are converted to thermal energy in 2.00 hours.
To solve this question, we need to find the amount of energy converted to thermal energy in 2.00 hours.
First, we need to calculate the power dissipated by the wire using the formula P = IV, where P is the power, I is the current, and V is the voltage.
The current can be found by substituting the given expression for current density J and the radius r into the formula I = ∫ J · dA, where I is the current and dA is the differential area element. Since the wire is circular, the differential area element can be expressed as dA = 2πr · dr. Thus, the current becomes I = ∫ J · 2πr · dr.
Using these expressions, we can calculate the current:
I = ∫ J · 2πr · dr
I = ∫ (2.75 × 10^10) · r^2 · 2πr · dr
I = 2.75 × 10^10 π ∫ r^3 · dr
I = 2.75 × 10^10 π · (1/4) r^4
I = (0.6875 × 10^10 π) r^4
Next, we can substitute the given values to calculate the power:
P = IV
P = (0.6875 × 10^10 π) r^4 · V
P = (0.6875 × 10^10 π) (3.00 × 10^-3)^4 · 60.0
Now that we have the power, we can calculate the amount of energy converted to thermal energy using the formula E = Pt, where E is the energy, P is the power, and t is the time:
E = Pt
E = [(0.6875 × 10^10 π) (3.00 × 10^-3)^4 · 60.0] · 2.00
Simplifying the expression, we can find the answer to the question using a calculator or software.