^{1,a)}, Kiyomitsu Suzuki

^{1}and Yasunori Ohkuma

^{2}

### Abstract

We describe a method of measuring the spatial structures of electric fields produced by charge distributions such as those on strip electrodes, small disk electrodes, and long double-plate electrodes. An electric-field sensor with high sensitivity to ac fields is fabricated for the measurement using a thin copper sheet. The reliability of the sensor is confirmed using a parallel-plate capacitor. The electric fields are oscillated at a frequency of 300 kHz to operate the electric-field sensor successfully. The structures of the measured fields coincide well with those of theoretical fields derived from Coulomb's law. When a dielectric is inserted in an electric field, polarization charges appear on the surface of the dielectric and modify the electric field in empty space. We measure the modified field and confirm the well-known linear relation between the polarization of a dielectric and the electric field. Dielectric susceptibilities are estimated from the linear relation for four types of dielectric.

I. INTRODUCTION

II. ELECTRIC-FIELD SENSOR

III. EXPERIMENTAL SETUP

IV. MEASUREMENT OF ELECTRIC FIELDS

A. Parallel-plate capacitor

B. Charges on strip electrodes

C. Charges on small disk electrodes

D. Electric dipole

V. MEASUREMENT OF DIELECTRIC SUSCEPTIBILITY

VI. CONCLUSIONS

### Key Topics

- Electric fields
- 55.0
- Electrodes
- 50.0
- Dielectrics
- 34.0
- Electrical sensors
- 23.0
- Electric measurements
- 13.0

## Figures

Disk-type EF sensor consisting of conducting plates and guard rings. Charges induced on the conducting plates flow through resistor Rs . Voltage signals across Rs are observed using an oscilloscope.

Disk-type EF sensor consisting of conducting plates and guard rings. Charges induced on the conducting plates flow through resistor Rs . Voltage signals across Rs are observed using an oscilloscope.

(Color online) Top view of EF sensor. The conducting plate is electrically isolated from the guard ring (painted in black) using a thin plastic film. The signal cables between the conducting plates and terminals A and B are covered with an electrostatic shield.

(Color online) Top view of EF sensor. The conducting plate is electrically isolated from the guard ring (painted in black) using a thin plastic film. The signal cables between the conducting plates and terminals A and B are covered with an electrostatic shield.

Experimental setup. A potential with and f = 300 kHz is supplied between the p-electrode and n-electrode. Nodes C and D in the resistive voltage divider are used to measure V 0. Nodes E and F are used to measure the charge Qe on the p-electrode.

Experimental setup. A potential with and f = 300 kHz is supplied between the p-electrode and n-electrode. Nodes C and D in the resistive voltage divider are used to measure V 0. Nodes E and F are used to measure the charge Qe on the p-electrode.

Reciprocal distance dependence of the electric field in a parallel-plate capacitor. Closed circles and triangles are electric fields measured by EF sensors with and without guard rings, respectively. The solid line denotes the electric field computed from the potential of the electrodes.

Reciprocal distance dependence of the electric field in a parallel-plate capacitor. Closed circles and triangles are electric fields measured by EF sensors with and without guard rings, respectively. The solid line denotes the electric field computed from the potential of the electrodes.

Distance dependence of the electric field, normalized by its value at x = 0 on the central plane of a parallel-plate capacitor with d = 4 cm. The radius R of the electrode is marked, and the horizontal line at x = 8 cm denotes the diameter of the conducting plate in the EF sensor. The solid line denotes the electric field computed from the potential of the electrodes.

Distance dependence of the electric field, normalized by its value at x = 0 on the central plane of a parallel-plate capacitor with d = 4 cm. The radius R of the electrode is marked, and the horizontal line at x = 8 cm denotes the diameter of the conducting plate in the EF sensor. The solid line denotes the electric field computed from the potential of the electrodes.

Pair of strip electrodes used for generating linear charge distributions parallel to the z-axis. The rectangular-type EF sensor is installed at the origin along the z-axis to detect the y-component of the electric field.

Pair of strip electrodes used for generating linear charge distributions parallel to the z-axis. The rectangular-type EF sensor is installed at the origin along the z-axis to detect the y-component of the electric field.

Reciprocal distance dependence of the electric field divided by the linear charge density of the p-electrode. Closed circles are obtained from the measured values Ey and λ. The solid and dashed lines are calculated from Eqs. (6) and (7) , respectively.

Reciprocal distance dependence of the electric field divided by the linear charge density of the p-electrode. Closed circles are obtained from the measured values Ey and λ. The solid and dashed lines are calculated from Eqs. (6) and (7) , respectively.

Pair of small disk electrodes with surface charge densities . A disk-type EF sensor installed between the electrodes detects the y-component of the electric field.

Pair of small disk electrodes with surface charge densities . A disk-type EF sensor installed between the electrodes detects the y-component of the electric field.

Electric field produced by small disk electrodes. The abscissa denotes the reciprocal square distance and the ordinate denotes the electric field divided by the charge Qe on the p-electrode. Closed circles denote measured values. The solid and dashed lines are calculated from Eqs. (8) and (9) , respectively.

Electric field produced by small disk electrodes. The abscissa denotes the reciprocal square distance and the ordinate denotes the electric field divided by the charge Qe on the p-electrode. Closed circles denote measured values. The solid and dashed lines are calculated from Eqs. (8) and (9) , respectively.

Two sets of long double-plate electrodes at , parallel with the z-axis, that are used for generating continuous linear distributions of dipole moments. The distance between the p-electrode and n-electrode is δ. A rectangular-type EF sensor for measuring Ey is installed along the z-axis.

Two sets of long double-plate electrodes at , parallel with the z-axis, that are used for generating continuous linear distributions of dipole moments. The distance between the p-electrode and n-electrode is δ. A rectangular-type EF sensor for measuring Ey is installed along the z-axis.

Sectional drawing of electrodes and EF sensors on the -plane at . The electrodes are rotated in the clockwise direction to measure the θ dependence of the electric field. We define on the -axis. The surfaces of the EF sensor are oriented to the -axis to obtain Ex and to the -axis to obtain Ey . Shaded areas between the electrodes denote the dielectric samples used in Sec. V .

Sectional drawing of electrodes and EF sensors on the -plane at . The electrodes are rotated in the clockwise direction to measure the θ dependence of the electric field. We define on the -axis. The surfaces of the EF sensor are oriented to the -axis to obtain Ex and to the -axis to obtain Ey . Shaded areas between the electrodes denote the dielectric samples used in Sec. V .

Distance dependences of electric fields produced by continuous linear distributions of dipole moments. The ordinate is the electric field divided by the charge on the p-electrode. Closed triangles and circles denote measured values of and , respectively. Solid lines are calculated from the right-hand side of Eq. (13) divided by Qe for Ex at and from Eq. (14) for Ey at .

Distance dependences of electric fields produced by continuous linear distributions of dipole moments. The ordinate is the electric field divided by the charge on the p-electrode. Closed triangles and circles denote measured values of and , respectively. Solid lines are calculated from the right-hand side of Eq. (13) divided by Qe for Ex at and from Eq. (14) for Ey at .

Azimuthal dependences of electric fields produced by continuous linear distributions of dipole moments. Closed triangles and circles denote measured values of and at , respectively. Solid lines are calculated from Eqs. (13) and (14) .

Azimuthal dependences of electric fields produced by continuous linear distributions of dipole moments. Closed triangles and circles denote measured values of and at , respectively. Solid lines are calculated from Eqs. (13) and (14) .

Linear relation between dielectric polarization P and electric field E. The dielectric samples are (a) neoprene rubber, (b) cushion rubber, (c) bakelite, and (d) acryl glass. Solid lines are fitting lines of data for each sample.

Linear relation between dielectric polarization P and electric field E. The dielectric samples are (a) neoprene rubber, (b) cushion rubber, (c) bakelite, and (d) acryl glass. Solid lines are fitting lines of data for each sample.

Dielectric susceptibility χ as a function of specific permittivity . Labels (a) through (d) refer to the samples in Fig. 14 . The solid line denotes the theoretical relation .

Dielectric susceptibility χ as a function of specific permittivity . Labels (a) through (d) refer to the samples in Fig. 14 . The solid line denotes the theoretical relation .

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