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Chapter 23, Problem 3P

A uniformly charged ring of radius 10.0 cm has a total charge of 75.0 μC. Find the electric field on the axis of the ring at (a) 1.00 cm, (b) 5.00 cm, (c) 30.0 cm, and (d) 100 cm from the center of the ring.

(a)

Expert Solution
Check Mark
To determine

The electric field on the axis of a ring at 1.00cm .

Answer to Problem 3P

The electric field on the axis of a ring at 1.00cm is 6.64×106N/C away from the center of the ring.

Explanation of Solution

Given info: The radius of the uniformly charged ring is 10.0cm and the total charge on the ring is 75.0μC .

From the Coulomb’s law the formula to calculate the electric field at any distance on the axis due to a uniformly charged ring.

Ex=Kqx(r2+x2)32 (1)

Here,

K is the proportionality constant.

r is the radius of the ring.

x is the distance of the electric field point from the center of the ring.

The proportionality constant K is,

K=14πε0

Here,

ε0 is the absolute electrical permeability.

Substitute 8.85×1012F/m for ε0 in the above equation.

K=14π(8.85×1012F/m)=8.991×109N

The charge on the ring is positive so the direction of the electric field is away from the center of the ring.

Substitute 1.00cm for x , 10.0cm for r , 8.991×109N for K and 75.0μC for q in the above equation.

Ex=(8.991×109N)(75.0μC)(1.00cm)((10.0cm)2+(1.00cm)2)32=(8.991×109N)(75.0μC)(106C1μC)(1.00cm)(102m1cm)(101cm2)32(102m1cm)3=0.6643×107N/C6.64×106N/C

Conclusion:

Therefore, the electric field on the axis of a ring at 1.00cm is 6.64×106N/C away from the center of the ring.

(b)

Expert Solution
Check Mark
To determine

The electric field on the axis of a ring at 5.00cm .

Answer to Problem 3P

The electric field on the axis of a ring at 5.00cm is 2.41×107N/C away from the center of the ring.

Explanation of Solution

Given info: The radius of the uniformly charged ring is 10.0cm and the total charge on the ring is 75.0μC .

The charge on the ring is positive so the direction of the electric field is away from the center of the ring.

Substitute 5.00cm for x , 10.0cm for r , 8.991×109N for K and 75.0μC for q in equation (1).

Ex=(8.991×109N)(75.0μC)(5.00cm)((10.0cm)2+(5.00cm)2)32=(8.991×109N)(75.0μC)(106C1μC)(5.00cm)(102m1cm)(125cm2)32(102m1cm)3=2.4125×107N/C2.41×107N/C

Conclusion:

Therefore, the electric field on the axis of a ring at 5.00cm is 2.41×107N/C away from the center.

(c)

Expert Solution
Check Mark
To determine

The electric field on the axis of a ring at 30.0cm .

Answer to Problem 3P

The electric field on the axis of a ring at 30.0cm is 6.39×106N/C away from the center of the ring.

Explanation of Solution

Given info: The radius of the uniformly charged ring is 10.0cm and the total charge on the ring is 75.0μC .

The charge on the ring is positive so the direction of the electric field is away from the center of the ring.

Substitute 30.0cm for x , 10.0cm for r , 8.991×109N for K and 75.0μC for q in the above equation.

Ex=(8.991×109N)(75.0μC)(30.0cm)((10.0cm)2+(30.0cm)2)32=(8.991×109N)(75.0μC)(106C1μC)(30.0cm)(102m1cm)(1000cm2)32(102m1cm)3=0.6397×107N/C6.39×106N/C

Conclusion:

Therefore, the electric field on the axis of a ring at 30.0cm is 6.39×106N/C away from the center.

(d)

Expert Solution
Check Mark
To determine

The electric field on the axis of a ring at 100cm .

Answer to Problem 3P

The electric field on the axis of a ring at 100cm is 6.64×105N/C away from the center of the ring.

Explanation of Solution

Given info: The radius of the uniformly charged ring is 10.0cm and the total charge on the ring is 75.0μC .

The charge on the ring is positive so the direction of the electric field is away from the center of the ring.

Substitute 100cm for x , 10.0cm for r , 8.991×109N for K and 75.0μC for q in the above equation.

Ex=(8.991×109N)(75.0μC)(100cm)((10.0cm)2+(100cm)2)32=(8.991×109N)(75.0μC)(106C1μC)(100cm)(102m1cm)(10100cm2)32(102m1cm)3=0.0664×107N/C=6.64×105N/C

Conclusion:

Therefore, the electric field on the axis of a ring at 100cm is 6.64×105N/C away from the center.

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Students have asked these similar questions
A uniformly charged ring of radius 10.0 cm has a total charge of 75.0 μC. Find the electric field on the axis of the ring at (a) 1.00 cm, (b) 5.00 cm, (c) 30.0 cm, and (d) 100 cm from the center of the ring.
A uniformly charged ring of radius 10.0 cm has a total charge of 47.0 μC. Find the electric field on the axis of the ring at the following distances from the center of the ring. (Choose the x-axis to point along the axis of the ring.) (a) 1.00 cm (b) 5.00 cm (c) 30.0 cm (d) 100 cm Î MN/C Î MN/C Î MN/C Î MN/C
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Chapter 23 Solutions

Bundle: Physics For Scientists And Engineers With Modern Physics, 10th + Webassign Printed Access Card For Serway/jewett's Physics For Scientists And Engineers, 10th, Multi-term

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