Question

A solid sphere of radius 40.0 cm has a total positive charge of 16.2 μC uniformly distributed throughout its volume. Calculate the magnitude of the electric field at the following distances.(a) 0 cm from the center of the sphere(b) 10.0 cm from the center of the sphere(c) 40.0 cm from the center of the sphere(d) 59.5 cm from the center of the sphere

Answer 1

Answer:

(a) E=0  :   0 cm from the center of the sphere

(b) E= 227.8*10³ N/C   :    10.0 cm from the center of the sphere

(c)E= 911.25*10³ N/C    :    40.0 cm from the center of the sphere

(d)E= 411.84 * 10³ N/C  :    59.5 cm from the center of the sphere

Explanation:

If we have a uniform charge sphere we can use the following formulas to calculate the Electric field due to the charge of the sphere

$E=\frac{K*Q}{r^{2} }$ : Formula (1) To calculate the electric field in the region outside the sphere r ≥ a

$E=k*\frac{Q}{a^{3} } *r$ :Formula (2) To calculate the electric field in the inner region of the sphere. r ≤ a

Where:

K: coulomb constant

a: sphere radius

Q:  Total sphere charge

r : Distance from the center of the sphere to the region where the electric field is calculated

Equivalences

1μC=10⁻⁶C

1cm= 10⁻²m

Data

k= 9*10⁹ N*m²/C²

Q=16.2 μC=16.2 *10⁻⁶C

a= 40 cm = 40*10⁻²m = 0.4m

Problem development

(a)Magnitude of the electric field at  0 cm :

We replace r=0 in the formula (2) , then, E=0

(b) Magnitude of the electric field at 10.0 cm from the center of the sphere

r<a , We apply the Formula (2):

$E=9*10^{9} *\frac{16.2*10^{-6} }{0.4^{3} } *0.1$

E= 227.8*10³ N/C

(c) Magnitude of the electric field at 40.0 cm from the center of the sphere

r=a, We apply the Formula (1) :

$E=\frac{9*10^{9}*16.2*10^{-6} }{0.4^{2} }$

E= 911.25*10³ N/C

(d) Magnitude of the electric field at 59.5 cm from the center of the sphere  

r>a , We apply the Formula (1) :

$E=\frac{9*10^{9}*16.2*10^{-6} }{0.595^{2} }$

E= 411.84 * 10³ N/C

Answer 2

(a) The magnitude of the electric field at a distance of 0 cm from the center of the sphere is 0.

(b) The magnitude of the electric field at a distance of 10 cm from the center of the sphere is 227,812.5 N/C.

(c) The magnitude of the electric field at a distance of 40 cm from the center of the sphere is 911,250 N/C.

(d) The magnitude of the electric field at a distance of 59.5 cm from the center of the sphere is 411,835.32 N/C.

Magnitude of electric field for region inside the sphere

The magnitude of electric field at any given position inside the sphere is calculated using the following formulas;

E = kQr/R³

When the distance , r = 0 cm

E = (kQ x 0)/R³ = 0

When the distance, r = 10 cm

E = (9 x 10⁹ x 16.2 x 10⁻⁶ x 0.1)/(0.4)³

E = 227,812.5 N/C

When the distance, r = 40 cm

E = KQr/R³

r = R

E = kQ/R²

E = (9 x 10⁹ x 16.2 x 10⁻⁶) / (0.4)²

E = 911,250 N/C

Magnitude of electric field for region outside the sphere

The magnitude of electric field at any given position outside the sphere is calculated using the following formulas;

E = kQ/R²

E = (9 x 10⁹ x 16.2 x 10⁻⁶)/(0.595)²

E = 411,835.32 N/C

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