To find the amount of heat produced when a dielectric is inserted into a capacitor, we first need to understand how the insertion of the dielectric affects the capacitor's charge and energy storage. Let's break this down step by step.
Understanding the Capacitor's Initial Conditions
Initially, we have a capacitor with a capacitance of 2 microfarads (μF) that stores a charge of 60 microcoulombs (μC). The relationship between charge (Q), capacitance (C), and voltage (V) is given by the formula:
Q = C × V
From this, we can find the initial voltage across the capacitor:
V = Q / C = 60 μC / 2 μF = 30 volts
Effect of Inserting the Dielectric
When a dielectric material with a dielectric constant (k) of 2 is inserted between the plates of the capacitor, the capacitance increases. The new capacitance (C') can be calculated as:
C' = k × C = 2 × 2 μF = 4 μF
With the dielectric in place, the charge on the capacitor increases to 120 microcoulombs (μC) as stated. This means that the battery must supply additional charge to the capacitor.
Calculating the New Voltage
Now, we can find the new voltage across the capacitor after the dielectric is inserted:
V' = Q' / C' = 120 μC / 4 μF = 30 volts
Interestingly, the voltage remains the same at 30 volts, which indicates that the battery maintains the voltage constant while the dielectric is inserted.
Energy Stored in the Capacitor
The energy stored in a capacitor is given by the formula:
U = 1/2 × C × V²
Calculating the initial energy (U_initial) before the dielectric is inserted:
U_initial = 1/2 × 2 μF × (30 V)² = 0.9 millijoules (mJ)
Now, calculating the energy (U_final) after the dielectric is inserted:
U_final = 1/2 × 4 μF × (30 V)² = 1.8 millijoules (mJ)
Heat Produced Due to Dielectric Insertion
The heat produced (Q_heat) when the dielectric is inserted can be determined by the change in energy stored in the capacitor:
Q_heat = U_final - U_initial = 1.8 mJ - 0.9 mJ = 0.9 millijoules (mJ)
Summary of Results
In conclusion, when the dielectric is inserted into the capacitor, the amount of heat produced is:
0.9 millijoules (mJ)
This heat is a result of the work done by the battery to move additional charge onto the capacitor plates, reflecting the energy change in the system.
