An empty capacitor has a capacitance of 1.86 μF and is connected to a 12-V battery. A dielectric material (κ = 4.5) is inserted between the plates of this capacitor. What is the magnitude of the surface charge on the dielectric that is adjacent to either plate of the capacitor? (Hint: The surface charge is equal to the difference in charge on the plates with and without the dielectric.)
The hint tells you. All you have to do it calculate capacitane with the dielerctric, then q on the plate with/without deelectric.
If the electric susceptibility κ=4.5,
electric permittivity is ε = κ+1 = 5.5
C₁=q₁/V,
q₁=C₁V=1.86•10⁻⁶•12=2.23•10⁻⁵ C,
C₁=ε₀A/d = q₁/V,
C₂=εε₀A/d = q₂/V,
q₂=εq₁ = 5.5•2.23•10⁻⁵ =1.23•10⁻⁴ C.
To find the magnitude of the surface charge on the dielectric, we first need to understand the relationship between capacitance, charge, and voltage, as well as how the presence of a dielectric affects these quantities.
1. Capacitance (C): Capacitance is a measure of a capacitor's ability to store charge. It is given by the formula C = Q/V, where Q is the charge stored on the capacitor and V is the voltage across the capacitor.
2. Charge (Q): Charge is the quantity of electric charge stored on a capacitor. In this case, we are interested in the difference in charge on the plates with and without the dielectric.
3. Voltage (V): Voltage is the potential difference across a capacitor. In this case, the capacitor is connected to a 12-V battery.
4. Dielectric Material: The introduction of a dielectric material between the plates of a capacitor increases its capacitance. The dielectric constant (κ) is a measure of how much the dielectric material increases the capacitance.
Now, let's calculate the surface charge on the dielectric adjacent to either plate of the capacitor:
Step 1: Without the dielectric
- The capacitor initially has a capacitance of 1.86 μF.
- Given that the voltage is 12 V, we can use the formula C = Q/V to find the charge on the capacitor without the dielectric.
- Rearranging the formula, Q = C * V, we get Q1 = (1.86 μF) * (12 V).
- Calculate Q1: Q1 = 1.86 * 12 = 22.32 μC (microcoulombs).
Step 2: With the dielectric
- The dielectric constant (κ) for this problem is given as 4.5.
- The capacitance of the capacitor with the dielectric can be calculated using the formula C' = κ * C, where C is the original capacitance.
- Calculate C': C' = 4.5 * (1.86 μF) = 8.37 μF.
- Using the formula C' = Q'/V, we can find the charge on the capacitor with the dielectric.
- Rearranging the formula, Q' = C' * V, we get Q2 = (8.37 μF) * (12 V).
- Calculate Q2: Q2 = 8.37 * 12 = 100.44 μC.
Step 3: Calculate the difference in charge
- The magnitude of the surface charge on the dielectric is equal to the difference in charge on the plates with and without the dielectric.
- Subtracting the charge without the dielectric from the charge with the dielectric, we get Q_diff = Q2 - Q1.
- Calculate Q_diff: Q_diff = 100.44 μC - 22.32 μC = 78.12 μC.
Therefore, the magnitude of the surface charge on the dielectric that is adjacent to either plate of the capacitor is 78.12 μC.