(1)A parallel-plate capacitor carrying charge q0 is modified by the insertion of a dielectric with κ = 4.3 between the plates. As a consequence, the energy stored in the capacitor quadruples. What will the charge be after the dielectric is inserted?
(2)A metal sphere of diameter 29 cm carries a charge of 5.6 10-7 C. How much energy is contained in a spherical region of radius 50 cm that is concentric with the sphere?
1) why would charge change? Isn't the capacitor isolated, with no where for the charge to go?
2) Energy= volume*energy density
where energy density is 1/2 epsilon*E^2
so an integration is required
E=kQ/r^2 dVolume= 4PIr^2 dr
energy= int 4PIr^2 (kQ/r^2)^2 dr
To solve these questions, we can use the basic formulas relating to capacitors and electric potential energy.
(1) The energy stored in a parallel-plate capacitor can be calculated using the formula:
U = (1/2) * C * V^2
Where U is the energy stored, C is the capacitance, and V is the potential difference.
Since the energy stored quadruples, we can write:
4U = (1/2) * C * V^2
Since the capacitance of a parallel-plate capacitor with a dielectric is given by:
C = κ * C0
Where C0 is the capacitance without the dielectric and κ is the dielectric constant, we can substitute this into the equation:
4U = (1/2) * κ * C0 * V^2
The charge q of the capacitor can be related to the capacitance and potential difference:
q = C * V
Substituting for C in terms of C0 and κ:
q = κ * C0 * V
Substituting this into our energy equation:
4U = (1/2) * q * V
Rearranging the equation for charge q:
q = 8U/V
Now we can use this equation to find q after the dielectric is inserted. Since the energy U is given for the initial capacitor, and the potential difference V remains the same, we can simply plug in the values into the equation to get the charge after the dielectric is inserted.
(2) The electric potential energy stored within a charged sphere can be calculated using the formula:
U = (3/5) * (q^2 / (4πε₀r))
Where U is the energy stored, q is the charge, r is the radius, and ε₀ is the permittivity of free space.
For a spherical region of radius R inside the larger sphere, the energy contained can be calculated as the difference between the energy of the larger sphere and the energy of the smaller sphere.
U_region = U_R - U_r
Substituting the formula for U into this equation:
U_region = (3/5) * [(q_R^2 / (4πε₀R)) - (q_r^2 / (4πε₀r))]
Where q_R is the charge of the larger sphere, q_r is the charge of the smaller sphere, R is the radius of the larger sphere, and r is the radius of the smaller sphere.
Now we can simply plug in the given values to calculate the energy contained in the specified region.