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A very long insulating cylinder of charge of radius 2.50cm carries a uniform linear density of 15.0nC/m. If you put one probe of a voltmeter at the surface, how far from the surface must the other probe be placed so that the voltmeter reads 175V.

Short Answer

Expert verified

The second probe should be placed at a distance of 2.3 cm from the surface of the cylinder to make the voltmeter measure 175 V.

Step by step solution

01

Potential law for infinite wire

Let a and b be two points, and the potential at those points is Va and Vb, respectively. The potential difference between two points for an infinite wire can be written as:

Va-Vb=λ2πε0lnrbra

Here ra and rb are the distance of the wire from points a and b, respectively.λis the linear chargedensity distributed across the wire.


02

Determine the distance between two probes of the voltmeter

The potential difference between two points due to a charged cylinder is given by:

ΔV=Va-Vb=λ2πε0lnrbra

Where raand rbare the distances between the cylinder's axis and the point where the potential is measured.

Here, ra=R=0.025mand the term 12πε0=18.0×109N·m2/C2

The target now is to get rbwhere the distance ra, in the problem equals the radius of the cylinder ra=R=0.025m. And the term 12πε0=18.0×109N·m2/C2. But, the distance rbis the distance between the axis of the cylinder and the point b, so the distance between the two probes of the voltmeter will be,

d=rb-R

03

Determine the distance where the second probe should be placed

Plug the values into the equation, and we get,

ΔV=λ2πε0lnrbra175V=18.0×109N·m2/C215.0×10-9C/mlnrb0.025mlnrb0.025m=0.648rb=0.025m×e0.648=0.048m

Now, plug this value to get the distance as:

d=rb-R=4.8cm-2.5cm=2.3cm

Thus, the second probe should be placed at a distance of 2.3 cm from the surface of the cylinder to make the voltmeter measure 175 V.

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