A uniformly charged disk of radius 35.0 cm carries a charge density of 6.70 multiplied by 10-3 C/m2. Calculate the electric field on the axis of the disk at the following distances from the center of the disk.

(a) 5.00 cm

Take the ring of the radius ‘r’ and the width ‘dr’ on the disc.

The electric field at the distance ‘x’ from the center of the disc is
dE=x•dq/4πε₀• {sqrt(r²+x²)}³,
where dq=σ•dA = σ•2•π•r•dr.
E(x)=
=intergral(limits: from 0 to R)
{σ•2•π•r•x•dr/ 4πε₀• [sqrt(r²+x²)]³ =
=(σx/2ε₀)•{(1/x)- [1/sqrt(R²+x²)]}.

weiners

To calculate the electric field on the axis of the disk at a certain distance, we can use the equation for the electric field due to a uniformly charged disk, given by:

E = (σ / 2ε0) * [1 - (z / √(z^2 + R^2))]

Where:
E is the electric field
σ is the charge density of the disk
ε0 is the permittivity of free space
z is the distance from the center of the disk
R is the radius of the disk

Now let's substitute the given values into the equation:

σ = 6.70 × 10^-3 C/m^2
R = 35.0 cm = 0.35 m
z = 5.00 cm = 0.05 m

Plugging in the values, we get:

E = (6.70 × 10^-3 C/m^2 / 2ε0) * [1 - (0.05 m / √(0.05 m^2 + 0.35 m^2))]

To simplify the equation further, we need the value of ε0, which is approximately 8.854 × 10^-12 C^2/(N·m^2).

Now let's calculate the electric field at z = 5.00 cm.

To calculate the electric field on the axis of the disk at a distance of 5.00 cm from the center, we can use the formula for the electric field of a uniformly charged disk.

The electric field at a point on the axis of a uniformly charged disk is given by:

E = (σ / (2ε₀)) * (z / R^2 + z^2)^(-3/2)

where:
E is the electric field,
σ is the charge density of the disk (in C/m^2),
ε₀ is the permittivity of free space (ε₀ ≈ 8.854 × 10^-12 C^2/Nm^2),
z is the distance from the center of the disk to the point on the axis (in m),
R is the radius of the disk (in m).

Given values:
σ = 6.70 × 10^-3 C/m^2
z = 5.00 cm = 0.05 m
R = 35.0 cm = 0.35 m

Substituting these values into the formula, we get:

E = (6.70 × 10^-3 C/m^2 / (2 × 8.854 × 10^-12 C^2/Nm^2)) * (0.05 m / (0.35 m^2 + 0.05 m^2))^(-3/2)

Simplifying further, we have:

E = (6.70 × 10^-3 C/m^2 / (2 × 8.854 × 10^-12 C^2/Nm^2)) * (0.05 m / (0.36 m^2))^(-3/2)

Calculating the denominator:

E = (6.70 × 10^-3 C/m^2 / (2 × 8.854 × 10^-12 C^2/Nm^2)) * (0.05 m / 0.1296 m^2)^(-3/2)

Simplifying the exponent:

E = (6.70 × 10^-3 C/m^2 / (2 × 8.854 × 10^-12 C^2/Nm^2)) * (0.05 m / 0.0114 m^2)^(3/2)

Calculating the rightmost term:

E = (6.70 × 10^-3 C/m^2 / (2 × 8.854 × 10^-12 C^2/Nm^2)) * (0.05 m / 0.004125 m)

Finally, we calculate the electric field E:

E ≈ (6.70 × 10^-3 C/m^2 / (2 × 8.854 × 10^-12 C^2/Nm^2)) * 12.12 N/C

E ≈ 510.2 N/C

Therefore, the electric field on the axis of the disk at a distance of 5.00 cm from the center is approximately 510.2 N/C.