Question

Two isolated, concentric, conducting spherical shells have radii R1 = 0.500 m and R2 = 1.00 m, uniform charges q1=+2.00 µC and q2 = +1.00 µC, and negligible thicknesses. What is the magnitude of the electric field E at radial distance (a) r = 4.00 m, (b) r =0.700 m, and (c) r = 0.200 m? With V = 0 at infinity, what is V at (d) r = 4.00 m, (e) r = 1.00 m, (f) r = 0.700 m, (g) r = 0.500 m, (h) r = 0.200 m, and (i) r = 0? (j) Plot the E(r) and V(r) dependencies.

Answer 1

Complete Question

The diagram for this question is shown on the first uploaded image

Answer:

a E =
`1.685*10^3 N/C`

b E =
`36.69*10^3 N/C`

c E = 0 N/C

d
`V = 6.7*10^3 V`

e
`V = 26.79*10^3V`

f V =
`34.67 *10^3 V`

g
`V= 44.95*10^3 V`

h
`V= 44.95*10^3 V`

i
`V= 44.95*10^3 V`

Step-by-step explanation:

From the question we are given that

The first charge
`q_1 = 2.00 \mu C = 2.00*10^(-6) C`

The second charge
`q_2 =1.00 \muC = 1.00*10^(-6)`

The first radius
`R_1 = 0.500m`

The second radius
`R_2 = 1.00m`

`Generally \ Electric \ field = (1)/(4\pi\epsilon_0)(q_1+\ q_2)/(r^2)`

And
`Potential \ Difference = (1)/(4\pi \epsilon_0) [(q_1 )/(r)+(q_2)/(R_2) ]`

The objective is to obtain the the magnitude of electric for different cases

And the potential difference for other cases

Considering a

r = 4.00 m

`E = (((2+1)*10^(-6))*8.99*10^9)/(16)`

`= 1.685*10^3 N/C`

Considering b

`r = 0.700 m \ , R_2 > r > R_1`

This implies that the electric field would be

`E = (1)/(4\pi \epsilon_0)(q_1)/(r^2)`

This because it the electric filed of the charge which is below it in distance that it would feel

`E = 8*99*10^9 (2*10^(-6))/(0.4900)`

=
`36.69*10^3 N/C`

Considering c

r = 0.200 m

=>
`r<R_1<R_2`

The electric field = 0

This is because the both charge are above it in terms of distance so it wont feel the effect of their electric field

Considering d

r = 4.00 m

=>
`r > R_1 >r>R_2`

Now the potential difference is

`V =(1)/(4\pi \epsilon_0) (q_1 + \ q_2)/(r) = 8.99*10^9 * (3*10^(-6))/(4) = 6.7*10^3 V`

This so because the distance between the charge we are considering is further than the two charges given

Considering e

r = 1.00 m
`R_2 = r > R_1`

`V = (1)/(4\pi \epsilon_0) [(q_1)/(r) +(q_2)/(R_2) ] = 8.99*10^9 * [(2.00*10^(-6))/(1.00) (1.00*10^(-6))/(1.00) ] = 26.79 *10^3 V`

Considering f

`r = 0.700 m \ , R_2 > r > R_1`

`V = (1)/(4\pi \epsilon_0) [(q_1)/(r) +(q_2)/(R_2) ] = 8.99*10^9 * [(2.00*10^(-6))/(0.700) (1.0*10^(-6))/(1.00) ] = 34.67 *10^3 V`

Considering g

`r =0.500\m , R_1 >r =R_1`

`V = (1)/(4\pi \epsilon_0) [(q_1)/(r) +(q_2)/(R_2) ] = 8.99*10^9 * [(2.00*10^(-6))/(0.500) (1.0*10^(-6))/(1.00) ] = 44.95 *10^3 V`

Considering h

`r =0.200\m , R_1 >R_1>r`

`V = (1)/(4\pi \epsilon_0) [(q_1)/(R_1) +(q_2)/(R_2) ] = 8.99*10^9 * [(2.00*10^(-6))/(0.500) (1.0*10^(-6))/(1.00) ] = 44.95 *10^3 V`

Considering i

`r =0\ m \ , R_1 >R_1>r`

`V = (1)/(4\pi \epsilon_0) [(q_1)/(R_1) +(q_2)/(R_2) ] = 8.99*10^9 * [(2.00*10^(-6))/(0.500) (1.0*10^(-6))/(1.00) ] = 44.95 *10^3 V`