A Geiger–Mueller tube is a radiation detector that consists of a closed, hollow, metal cylinder (the cathode) of inner radius r a and a coaxial cylindrical wire (the anode) of radius r b (Fig. P24.42a). The charge per unit length on the anode is λ , and the charge per unit length on the cathode is − λ . A gas fills the space between the electrodes. When the tube is in use (for example, in measuring radioactivity from fruit in Fig. P24.42b) and a high-energy elementary particle passes through this space, it can ionize an atom of the gas. The strong electric field makes the resulting ion and electron accelerate in opposite directions. They strike other molecules of the gas to ionize them, producing an avalanche of electrical discharge. The pulse of electric current between the wire and the cylinder is counted by an external circuit. (a) Show that the magnitude of the electric potential difference between the wire and the cylinder is Δ V − 2 k e λ ln ( r a r b ) (b) Show that the magnitude of the electric field in the space between cathode and anode is E = Δ V ln ( r a / r b ) ( 1 r ) where r is the distance from the axis of the anode to the point where the field is to be calculated. Figure P24.42
A Geiger–Mueller tube is a radiation detector that consists of a closed, hollow, metal cylinder (the cathode) of inner radius r a and a coaxial cylindrical wire (the anode) of radius r b (Fig. P24.42a). The charge per unit length on the anode is λ , and the charge per unit length on the cathode is − λ . A gas fills the space between the electrodes. When the tube is in use (for example, in measuring radioactivity from fruit in Fig. P24.42b) and a high-energy elementary particle passes through this space, it can ionize an atom of the gas. The strong electric field makes the resulting ion and electron accelerate in opposite directions. They strike other molecules of the gas to ionize them, producing an avalanche of electrical discharge. The pulse of electric current between the wire and the cylinder is counted by an external circuit. (a) Show that the magnitude of the electric potential difference between the wire and the cylinder is Δ V − 2 k e λ ln ( r a r b ) (b) Show that the magnitude of the electric field in the space between cathode and anode is E = Δ V ln ( r a / r b ) ( 1 r ) where r is the distance from the axis of the anode to the point where the field is to be calculated. Figure P24.42
A Geiger–Mueller tube is a radiation detector that consists of a closed, hollow, metal cylinder (the cathode) of inner radius ra and a coaxial cylindrical wire (the anode) of radius rb (Fig. P24.42a). The charge per unit length on the anode is λ, and the charge per unit length on the cathode is −λ. A gas fills the space between the electrodes. When the tube is in use (for example, in measuring radioactivity from fruit in Fig. P24.42b) and a high-energy elementary particle passes through this space, it can ionize an atom of the gas. The strong electric field makes the resulting ion and electron accelerate in opposite directions. They strike other molecules of the gas to ionize them, producing an avalanche of electrical discharge. The pulse of electric current between the wire and the cylinder is counted by an external circuit. (a) Show that the magnitude of the electric potential difference between the wire and the cylinder is
Δ
V
−
2
k
e
λ
ln
(
r
a
r
b
)
(b) Show that the magnitude of the electric field in the space between cathode and anode is
E
=
Δ
V
ln
(
r
a
/
r
b
)
(
1
r
)
where r is the distance from the axis of the anode to the point where the field is to be calculated.
(a) Find the electric field at x = 5.00 cm in (a), given that q = 1.00 μC . (b) At what position between 3.00 and 8.00 cm is the total electric field the same as that for –2q alone? (c) Can the electric field be zero anywhere between 0.00 and 8.00 cm? (d) At very large positive or negative values of x, the electric field approaches zero in both (a) and (b). In which does it most rapidlyapproach zero and why? (e) At what position to the right of 11.0 cm is the total electric field zero, other than at infinity?(Hint: A graphing calculator can yield considerable insight in this problem.)
The figure below is a section of a conducting rod of radius R1 = 1.30 mm and length L = 11.00 m inside a thick-walled coaxial conducting cylindrical shell of radius R2 = 10.0R1 and the (same) length L. The net charge on the rod is Q1 = +4.60 ✕ 10−12 C; that on the shell is Q2 = −4.00Q1.
(a) What is the magnitude E of the electric field at a radial distance of r = 3.50R2? _________N/C(b) What is the direction of the electric field at that radial distance? ---Select---: inward or outward(c) What is the magnitude E of the electric field at a radial distance of r = 3.50R1? ___________N/C
(a) Using the symmetry of the arrangement, determine the direction of the electric field at the center of the square in Figure 18.53, given that qa=qb=1.00 μC and qc=qd=+1.00 μC . (b) Calculate the magnitude of the electric field at the location of q , given that the square is 5.00 cm on a side.
Chapter 25 Solutions
Physics for Scientists and Engineers, Technology Update (No access codes included)
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