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mr_godi [17]
3 years ago
9

A spherical capacitor contains a charge of 3.00 nC when connected to a potential difference of 230 V. If its plates are separate

d by vacuum and the inner radius of the outer shell is 4.30 cm.
Calculate: (a) the capacitance; (b) the radius of the inner sphere; (c) the electric field just outside the surface of the inner sphere.
Physics
1 answer:
Assoli18 [71]3 years ago
3 0

Answer:

Part(a): the capacitance is 0.013 nF.

Part(b): the radius of the inner sphere is 3.1 cm.

Part(c): the electric field just outside the surface of inner sphere is \bf{2.81 \times 10^{4}~n~C^{-1}}.

Explanation:

We know that if 'a' and 'b' are the inner and outer radii of the shell respectively, 'Q' is the total charge contains by the capacitor subjected to a potential difference of 'V' and '\epsilon_{0}' be the permittivity of free space, then the capacitance (C) of the spherical shell can be written as

C = \dfrac{4 \pi \epsilon_{0}}{(\dfrac{1}{a} - \dfrac{1}{b})}~~~~~~~~~~~~~~~~~~~~~~~~~~~(1)

Part(a):

Given, charge contained by the capacitor Q = 3.00 nC and potential to which it is subjected to is V = 230V.

So the capacitance (C) of the shell is

C &=& \dfrac{Q}{V} = \dfrac{3 \times 10^{-90}~C}{230~V} = 1.3 \times 10^{-11}~F = 0.013~nF

Part(b):

Given the inner radius of the outer shell b = 4.3 cm = 0.043 m. Therefore, from equation (1), rearranging the terms,

&& \dfrac{1}{a} = \dfrac{1}{b} + \dfrac{1}{C/4 \pi \epsilon_{0}} = \dfrac{1}{0.043} + \dfrac{1}{1.3 \times 10^{-11} \times 9 \times 10^{9}} = 31.79\\&or,& a = \dfrac{1}{31.79}~m = 0.031~m = 3.1~cm

Part(c):

If we apply Gauss' law of electrostatics, then

&& E~4 \pi a^{2} = \dfrac{Q}{\epsilon_{0}}\\&or,& E = \dfrac{Q}{4 \pi \epsilon_{0}a^{2}}\\&or,& E = \dfrac{3 \times 10^{-9} \times 9 \times 10^{9}}{0.031^{2}}~N~C^{-1}\\&or,& E = 2.81 \times 10^{4}~N~C^{-1}

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A bicycle wheel has a diameter of 63.0 cm and a mass of 1.75 kg. Assume that the wheel is a hoop with all of the mass concentrat
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Answer:

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Diameter; d2 = 8.96 cm = 0.0896 m

Radius; r2 = 0.0896/2 = 0.0448 m

Radial acceleration; α = 4.4 rad/s²

Now moment of inertia of the wheel which is assumed to be a hoop is given by; I = m(r1)²

Where r1 = (d1)/2 = 0.63/2

r1 = 0.315 m

Thus, I = 1.75 × 0.315²

I = 0.1736 Kg.m²

The torque is given by the relation;

I•α = F1•r1 - F2•r2

Where F2 is the force that must be applied by the chain to give the wheel an acceleration of 4.40 rad/s².

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0.1736 × 4.4 = (121 × 0.315) - (0.0448F2)

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A 10-cm-long thin glass rod uniformly charged to 8.00 nCnC and a 10-cm-long thin plastic rod uniformly charged to - 8.00 nCnC ar
ch4aika [34]

Complete Question

A 10-cm-long thin glass rod uniformly charged to 8.00 nC and a 10-cm

long thin plastic rod uniformly charged to -8.00 nC are placed side by

side, 4.20 cm apart. What are the electric field strengths E_1 to E_3 at

distances 1.0 cm, 2.0 cm, and 3.0 cm from the glass rod along the line

connecting the midpoints of the two rods

a.) Specify the electric field strength E1

b.) Specify the electric field strength E2

c.) Specify the electric field strength E3

Answer:

              E_1=7.13*10^5 N/C

             E_2= 2.95*10^{5} N/C

              E_3= 3.84*10^5 N/C

Explanation:

  From the question we are told that

          The length of the thin glass is  L = 10 cm

          The  charge on the glass rod is  q_g = 8.00nC = 8* 10^{-9} C

           The length of the plastic rod is  L_p = 10cm

             The charge on the  plastic rod is q_p =- 8.00nC = -8.0*10^{-9}C

           The distance between the materials  is d = 4.20cm = \frac{4.2}{100} =0.042m

          The various distances to obtain electric field of are r_1 = 1.0cm

                                                                                                r_2 = 2.0cm

                                                                                                 r_3 = 3.0cm

The objective of the solution is to obtain the electric field E_1 , E_2 \ and E_3 at distance d_1 , d_2 \ and \ d_3  from the glass rod  along the line connecting its mid point  

   Generally electric field of a charge rod at a distance of r the line dividing the rod  into half  is mathematically represented as

                              E = k \frac{2Q}{r\sqrt{L^2 + 4r^2} }

For the  r_2 = 1.0cm = \frac{1}{100} = 0.01m

The electric filed by the positively charge glass rod on the left side of the dividing line is mathematically represented as

                               E_l =  k \frac{2Q }{r \sqrt{L^2 + 4r^2_1} }

The electric filed by the positively charge glass rod on the right  side of the dividing line is mathematically represented as  

                            E_r =  k \frac{2Q }{(0.044 - r_1) \sqrt{L^2 + 4r^2_1} }

The net electric field is,

            E_{net} =E_1= E_l + E_r

                    = k \frac{2Q}{r_1\sqrt{L^2 + 4 r^2_1 } } + k \frac{2Q}{(0.04-r_1) \sqrt{L^2 + 4 (0.044 -r_1)^2} }

Where k is  know as the coulomb's constant  with a constant value of

                  k = 9*10^9 \ kgm^3 s^{-4} A^{-2}

           =(9*10^9) \frac{(2) (8*10^{-9})}{(0.01)\sqrt{(0.01^2 + 4(0.01)^2)} }  + (9* 10^9 ) \frac{(2)(8*10^{-9})}{(0.0420 - 0.01)\sqrt{(0.01)^2 + (4) (0.042 - 0.01)^2} }

                           = 6.44*10^5 + 6.9*10^4

                           E_1=7.13*10^5 N/C

For the  r_2 = 2.0cm = \frac{2}{100} = 0.02m

           The electric filed by the positively charge glass rod on the left side of the dividing line is mathematically represented as

                               E_l =  k \frac{2Q }{r_2 \sqrt{L^2 + 4r^2_2} }

The electric filed by the positively charge glass rod on the right  side of the dividing line is mathematically represented as  

                            E_r =  k \frac{2Q }{(0.044 - r_2) \sqrt{L^2 + 4r^2_2} }

The net electric field is,

            E_{net} =E_2= E_l + E_r

                    = k \frac{2Q}{r_2\sqrt{L^2 + 4 r^2_2 } } + k \frac{2Q}{(0.04-r_2) \sqrt{L^2 + 4 (0.044 -r_2)^2} }

Where k is  know as the coulomb's constant  with a constant value of

                  k = 9*10^9 \ kgm^3 s^{-4} A^{-2}

           =(9*10^9) \frac{(2) (8*10^{-9})}{(0.02)\sqrt{(0.02^2 + 4(0.02)^2)} }  + (9* 10^9 ) \frac{(2)(8*10^{-9})}{(0.0420 - 0.02)\sqrt{(0.02)^2 + (4) (0.042 - 0.02)^2} }

            = 1.6*10^{5}+ 1.3*10^{5}

             E_2= 2.95*10^{5} N/C

For the  r_3 = 3.0cm = \frac{3}{100} = 0.03m

           The electric filed by the positively charge glass rod on the left side of the dividing line is mathematically represented as

                               E_l =  k \frac{2Q }{r_3 \sqrt{L^2 + 4r^2_3} }

The electric filed by the positively charge glass rod on the right  side of the dividing line is mathematically represented as  

                            E_r =  k \frac{2Q }{(0.044 - r_3) \sqrt{L^2 + 4r^2_3} }

The net electric field is,

            E_{net} =E_3= E_l + E_r

                    = k \frac{2Q}{r_3\sqrt{L^2 + 4 r^2_3 } } + k \frac{2Q}{(0.04-r_3) \sqrt{L^2 + 4 (0.044 -r_3)^2} }

Where k is  know as the coulomb's constant  with a constant value of

                  k = 9*10^9 \ kgm^3 s^{-4} A^{-2}

           =(9*10^9) \frac{(2) (8*10^{-9})}{(0.03)\sqrt{(0.03^2 + 4(0.03)^2)} }  + (9* 10^9 ) \frac{(2)(8*10^{-9})}{(0.0420 - 0.03)\sqrt{(0.03)^2 + (4) (0.042 - 0.03)^2} }

        = 7.2 *10^{4} + 3.1*10^5

      E_3= 3.84*10^5 N/C                

8 0
3 years ago
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