A battery has an emf of 15.0 V. The terminal voltage of the battery is 11.2 V when it is delivering 26.0 W of power to an external load resistor R.
(a) What is the value of R?
(b) What is the internal resistance of the battery?
The volts across your load resistor = 11.2V
Power delivered = 26.0 W
Power = Volts x Amps so Current (Amps) = 26.0/11.2 V = 2.32A
(A) 11.2V/2.32A = 4.83 ohm
Volts lost across internal resistance = 15 - 11.2 = 3.8V
Internal resistance = Lost Volts/Current = 3.8/2.32 = 1.64 ohms (B)
(a) Well, let's do some math to find out the value of R. We know that power (P) is equal to the square of the voltage (V) divided by the resistance (R). So, we have P = V^2 / R.
Given that the terminal voltage (V) is 11.2 V and the power (P) is 26.0 W, we can rearrange the equation to solve for R:
R = V^2 / P
Substituting the values, we get R = (11.2 V)^2 / 26.0 W. Calculating this, we find that R is approximately 4.81 ohms.
(b) Now, let's get to the internal resistance. We can use Ohm's Law to find it. Ohm's Law states that V = I * R, where V is the voltage, I is the current, and R is the resistance.
In this case, we know that the emf (E) of the battery is 15.0 V, and the terminal voltage (V) is 11.2 V. The difference between these two voltages is due to the internal resistance (r). So, we have:
V = E - I * r
Substituting the values, we get 11.2 V = 15.0 V - I * r. Rearranging the equation, we find:
I * r = 15.0 V - 11.2 V
I * r = 3.8 V
Now, we also know that power (P) is equal to the current (I) multiplied by the voltage (V), so we have P = I * V. Substituting the power (P) value of 26.0 W and the voltage (V) value of 11.2 V, we can solve for the current (I):
26.0 W = I * 11.2 V
Solving for I, we find that I is approximately 2.32 A.
Now, let's substitute this value of I into the previous equation:
2.32 A * r = 3.8 V
Solving for r, we get:
r = 3.8 V / 2.32 A
Calculating this, we find that the internal resistance of the battery is approximately 1.64 ohms.
To find the value of the load resistor (R) and the internal resistance of the battery, we will use the formulas:
a) V = IR, where V is the voltage, I is the current, and R is the resistance.
b) P = IV, where P is the power, I is the current, and V is the voltage.
Given information:
Emf of the battery (E) = 15.0 V
Terminal voltage (V) = 11.2 V
Power (P) = 26.0 W
(a) To find the value of the load resistor (R):
Step 1: Use the formula V = IR to find the current (I).
11.2 V = I * R ----(1)
Step 2: Use the formula P = IV to find the current (I).
26.0 W = I * 11.2 V ----(2)
Step 3: Rearrange equation (2) to solve for I.
I = 26.0 W / 11.2 V
Step 4: Substitute the value of I in equation (1).
11.2 V = (26.0 W / 11.2V) * R
Step 5: Solve for R.
R = (11.2 V * 11.2V) / 26.0 W
(b) To find the internal resistance of the battery:
Step 1: Use the formula V = E - Ir, where V is the terminal voltage, E is the emf of the battery, I is the current, and r is the internal resistance.
11.2 V = 15.0 V - I * r
Step 2: Rearrange the equation to solve for r.
r = (15.0 V - 11.2 V) / I
Step 3: Substitute the value of I obtained from equation (2).
r = (15.0 V - 11.2 V) / (26.0 W / 11.2 V)
To find the value of R, we first need to understand the relationship between emf, terminal voltage, and power.
The emf (ε) of a battery represents the maximum potential difference it can provide to a circuit when no current is flowing. The terminal voltage (V) of the battery is the actual voltage measured across its terminals when current is flowing through a circuit. The power (P) delivered to a load resistor is given by P = V^2 / R, where R is the resistance of the load.
In this case, we are given the emf of the battery (ε = 15.0 V), the power delivered to the external load resistor (P = 26.0 W), and the terminal voltage (V = 11.2 V). We need to find the value of R.
To find the value of R, we can rearrange the power formula to solve for R:
R = V^2 / P
Plugging in the given values:
R = (11.2 V)^2 / 26.0 W
R = 125.44 V^2 / 26.0 W
R = 4.826 Ω
Therefore, the value of R is approximately 4.826 Ω.
To find the internal resistance of the battery, we can use the following equation:
V = ε - I * r
Where V is the terminal voltage, ε is the emf of the battery, I is the current flowing through the circuit, and r is the internal resistance of the battery.
In this case, we are given the terminal voltage (V = 11.2 V) and the emf of the battery (ε = 15.0 V). We don't have direct information about the current (I), so we need to solve for it.
We know that power (P) is given by P = V * I. Since we are given the power delivered to the external load resistor (P = 26.0 W), we can rewrite the power equation as I = P / V. Then, we can substitute this value of I in the equation for the terminal voltage:
V = ε - (P / V) * r
Now, we can rearrange the equation to solve for r:
r = (ε * V - P) / (V * I)
Plugging in the given values:
r = (15.0 V * 11.2 V - 26.0 W) / (11.2 V * (26.0 W / 11.2 V))
r = (168.0 V^2 - 26.0 W) / (26.0 W / 2.35)
r = (168.0 V^2 - 26.0 W) / 11.06 W^-1
r ≈ 14.9 Ω
Therefore, the internal resistance of the battery is approximately 14.9 Ω.