Neutral hydrogen can be modeled as a positive point charge + 1.6 × 10 −19 C surrounded by a distribution of negative charge with volume density given by ρ E ( r ) = − A e − 2 r / a 0 where a 0 = 0.53 × 10 −10 m is called the Bohr radius, A is a constant such that the total amount of negative charge is −1.6 × 10 −9 C, and e = 2.718 ⋯ is the base of the natural log. ( a ) What is the net charge inside a sphere of radius a 0 ? ( b ) What is the strength of the electric field at a distance a 0 from the nucleus? [ Hint : Do not confuse the exponential number e with the elementary charge e which uses the same symbol but has a completely different meaning and value ( e = 1.6 × 10 −19 C).]
Neutral hydrogen can be modeled as a positive point charge + 1.6 × 10 −19 C surrounded by a distribution of negative charge with volume density given by ρ E ( r ) = − A e − 2 r / a 0 where a 0 = 0.53 × 10 −10 m is called the Bohr radius, A is a constant such that the total amount of negative charge is −1.6 × 10 −9 C, and e = 2.718 ⋯ is the base of the natural log. ( a ) What is the net charge inside a sphere of radius a 0 ? ( b ) What is the strength of the electric field at a distance a 0 from the nucleus? [ Hint : Do not confuse the exponential number e with the elementary charge e which uses the same symbol but has a completely different meaning and value ( e = 1.6 × 10 −19 C).]
Neutral hydrogen can be modeled as a positive point charge + 1.6 × 10−19 C surrounded by a distribution of negative charge with volume density given by
ρ
E
(
r
)
=
−
A
e
−
2
r
/
a
0
where a0 = 0.53 × 10−10 m is called the Bohr radius, A is a constant such that the total amount of negative charge is −1.6 × 10−9 C, and
e
=
2.718
⋯
is the base of the natural log. (a) What is the net charge inside a sphere of radius a0? (b) What is the strength of the electric field at a distance a0 from the nucleus? [Hint: Do not confuse the exponential number e with the elementary charge e which uses the same symbol but has a completely different meaning and value (e = 1.6 × 10−19C).]
Suppose a capacitor consists of two coaxial thin cylindrical conductors. The inner cylinder of radius ra has a charge of +Q, while the outer cylinder of radius rp has charge -Q. The
electric field E at a radial distance r from the central axis is given by the function:
E = ae-T/ao + B/r + bo
where alpha (a), beta (8), ao and bo are constants. Find an expression for its capacitance.
First, let us derive the potential difference Vah between the two conductors. The potential difference is related to the electric field by:
Vab =
Edr = -
Edr
Calculating the antiderivative or indefinite integral,
Vab = (-aage-r/a0 + B
+ bo
By definition, the capacitance Cis related to the charge and potential difference by:
C =
Evaluating with the upper and lower limits of integration for Vab, then simplifying:
C= Q/(
(e-rb/ao - eTalao) + B In(
) + bo (
))
Suppose a capacitor consists of two coaxial thin cylindrical conductors. The inner cylinder of
radius ra has a charge of +Q, while the outer cylinder of radius rp has charge -Q. The electric field
E at a radial distance r from the central axis is given by the function:
E = aer/ao + B/r + bo
%3D
where alpha (a), beta (B), ao and bo are constants. Find an expression for its capacitance.
First, let us derive the potential difference Vab between the two conductors. The potential
difference is related to the electric field by:
Vab = |
S"Edr= - [ *Edr
Calculating the antiderivative or indefinite integral,
Vab = (-aage-r/ao + B
+ bo
By definition, the capacitance C is related to the charge and potential difference by:
C =
Evaluating with the upper and lower limits of integration for Vab, then simplifying:
C = Q/(
(e-rb/ao - eralao) + B In(
) + bo (
))
Suppose a capacitor consists of two coaxial thin cylindrical conductors. The inner cylinder of radius ra has a charge of +Q, while the outer cylinder of radius rh has charge -Q.
srb
The electric field E at a radial distance r from the central axis is given by the function:
E = ge/d0 + B/r + bo
where alpha (a)., beta (8), ao and bo are constants. Find an expression for its capacitance.
First, let us derive the potential difference Voh between the two conductors. The potential difference is related to the electric field by:
Edr = -
Edr
Calculating the antiderivative or indefinite integral,
Vab = (-aageao + B
+ bo
By definition, the capacitance Cis related to the charge and potential difference by:
C =
Evaluating with the upper and lower limits of integration for Vab, then simplifying:
C = Q/(
(e""b/ao - eTala0) + ß In
) + bo (
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